c 2005 Society for Industrial and Applied Mathematics

SIAM J. MATH. ANAL. Vol. 37, No. 1, pp. 17–59

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES∗ ´ † AND LAURENT YOUNES‡ ALAIN TROUVE Abstract. In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which inﬁnitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield eﬃcient and robust estimations of diﬀeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this inﬁnite dimensional “shape manifold” on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We ﬁnally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template. Key words. inﬁnite dimensional Riemannian manifolds, deformable templates, shape representation and recognition, warping AMS subject classiﬁcations. Primary, 58b10; Secondary, 49J45, 68T10 DOI. 10.1137/S0036141002404838

1. Introduction. The theoretical developments which are addressed in this paper are motivated by the theory of deformable templates, as it emerged from the work of Grenander and his collaborators in the 1980’s [19, 21, 22, 20], to handle image processing problems. This theory has an abstract formulation, in which the purpose is to represent the variability within an object class by the variations in shape, or color, etc., of a single object, submitted to the action of “deformations.” For instance, a model designed to describe a picture of a human face should be able to explain interindividual variations but also variations caused by the change of expression of a given individual, and by the changing of imaging conditions, such as lighting, occultations, etc. The interesting feature in Grenander’s construction is that it assigns a large part, sometimes all, of the variations to a ﬁxed structure, describing the deformation, which is independent of the particular instance of the observed image. This structure most of the time belongs to a group, the group of deformations, which is acting on the set of objects. The speciﬁc choice of the group depends on the application and on the type of visual features which are modeled, like pixelized images [18] and discretized shapes [20, 29, 23]. In such discrete settings, the group action is used to generate variations of the constituting generators of the object (pixels for an image, segments for polygons) and therefore are modeled as ﬁnite dimensional groups, generally products of linear or aﬃne groups. In the simple example of labeled collections of points (landmarks), the deformation may simply correspond to independent translations of each point, but when the question is raised of the similarity of two collections of landmarks, one would like to ﬁgure out the amount of deformation which is required to transform one of them into the other. When evaluating this deformation, it is clear that the lengths of the induced translations should have some impact, but that this is not the ∗ Received by the editors March 29, 2002; accepted for publication (in revised form) June 11, 2004; published electronically August 17, 2005. http://www.siam.org/journals/sima/37-1/40483.html † LAGA (UMR CNRS 7593), Institut Galil´ ee, Universit´e Paris XIII, Av. J-B. Cl´ ement, F-93430 Villetaneuse, France ([email protected]). ‡ CMLA (CNRS, URA 1611), Ecole Normale Sup´ erieure de Cachan, 61 avenue du Pr´esident Wilson, F-94235 Cachan Cedex, France ([email protected]).

17

18

´ AND LAURENT YOUNES ALAIN TROUVE

only factor and often not even the main factor. One would also like to draw conclusions on the smoothness of the deformation, based on the fact that, in the context of large deformations of shapes, a lower similarity must be associated to a collection of translations which point to erratic directions, compared to a more homogeneous displacement. We see, in this case, that a global point of view on the displacements is needed. Spline-based landmark matching [9, 26] speciﬁcally addresses this issue by seeking the smoothest function which interpolates the considered displacements. When dealing with image deformation, the need to pass to the continuum is even more obvious. In this case, deformations, which should provide nonambiguous point displacements, must be diﬀeomorphisms on the image support. This nonambiguity constraint, however, has been relaxed in most of the early attempts to deal with this issue, working preferably with linear spaces of deformations [6, 7, 8, 2, 1, 14], which can be seen as ﬁrst order approximations. Dealing explicitly with true deformations, i.e., diﬀeomorphisms acting on the support of images, was rigorously formalized by Riemannian metric arguments on the groups of diﬀeomorphisms in [32] for onedimensional problems, and in [31] in full generality (see also [30]). Stemming from the simple representation of right invariant metrics on groups of diﬀeomorphisms along a path in this space, i.e., time-dependent deformations, in terms of the Eulerian velocity, this last reference built diﬀeomorphisms as ﬂows associated to ODEs (a construction which was already present in [3]) and transferred the modeling eﬀort to the linear space of velocities, i.e., of vector ﬁelds deﬁned on the image support. Under suitable Banach space structures on these linear spaces, the extension of the ODE solutions for inﬁnite time and the existence of minimizers to general variational problems in this space can be ensured, providing rigorous suﬃcient conditions for the well-posedness of many practical problems in template matching. This analysis rejoined the line of work of Miller and his collaborators on the estimation of large deformation diﬀeomorphisms [13, 26], in which velocity-based models have been introduced, and variational properties studied in [16]. In [27], the interest in considering a lifted group action, on the cross product of the group itself and of the image space, was demonstrated in a wide variety of applications. The ﬁnal metric on the image space was obtained by projecting a right-invariant Riemannian distance designed on the product space. The approach we follow in this paper addresses the same kind of construction as in [27], which focused on the metric aspects, but from a diﬀerent point of view. Our purpose is to start from the inﬁnitesimal analysis of small deformations of images in order to model and measure image variations and deﬁne diﬀerentiable and geodesic curves in the image space. We shall accept conditions which ensure enough smoothness on the diﬀeomorphisms but try whenever possible to avoid placing such smoothness assumptions on the images themselves. Such a choice, which is very important given the discontinuous nature of images, is made at the cost of increasing technicalities and notation, as will be seen in section 3, in which the basic geometry of the model is presented. Here, we deﬁne the tangent space at a given square integrable image i as an equivalent class for all possible variations resulting from an inﬁnitesimal combination of a deformation (geometry) and of the addition of a square integrable function (photometry), yielding what can be called a morphometrical variation. We then equip it with an inner product and deﬁne from it lengths and energies of curves. This metric is based on the best tradeoﬀ between geometrical and photometrical variations. Still, in this general setting, we show the existence of minimizing geodesics (curves of minimal energy) between any two images. The rest of the paper is devoted to the study of geodesics and their generation

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

19

from initial conditions. The motivation in this study is the possibilities it oﬀers for prototype-based image representation and the generation of image variations and deformations from initial conditions belonging to a vector space. In this context, the geodesic equations are derived under the assumption that the deformed prototype is smooth (H 1 ), but with no restriction on the other endpoint. This is done in section 5.2.2. The obtained evolution equations are then generalized to a form which does not require the smoothness of the initial position and that we conjecture to represent a comprehensive class of image evolutions. The equations, under this form, are studied in section 7, where we prove that they have a unique solution over arbitrary ﬁnite time intervals. Our last result shows the local nonambiguity of this representation, at least in the smooth case: from a smooth prototype, the solutions of the geodesic equations in small time cannot coincide if they have been generated from distinct smooth initial conditions. This is done in section 9. The last section, 10, presents numerical experiments, which illustrate the feasibility of retrieving a target from the initial conditions associated to the minimizing geodesic starting from the template. 2. Notation. For further reference, we present in a single deﬁnition some of the main functional spaces we use throughout the paper. Definition 1. Let k, p ∈ N∗ , l ∈ N, and Ω be a bounded domain of Rk with C 1 boundary. (1) We denote Cc∞ (Ω, Rp ) the space of smooth compactly supported Rp -valued functions on Ω. (2) We denote C l (Ω, Rp ) the set of the restrictions to Ω of the l times continuously diﬀerentiable Rp -valued functions on Rk . Let f ∈ C l (Ω, Rp ). We deﬁne the norm |f |l,∞ by |f |l,∞

∂ |α| f , d · · · ∂xα d

sup |

α1 x∈Ω ∂x1 α, 0≤|α|≤l

where for any α (α1 , . . . , αd ) ∈ Nd∗ we denote |α| αi . (3) We denote C0l (Ω, Rp ) the completion of Cc∞ (Ω, Rp ) for the norm | |l,∞ . (4) We denote L2 (Ω, Rp ) the Hilbert space of square integrable functions in Rp with dot product deﬁned for f, g ∈ L2 (Ω, Rp ) by f, g2 f (x), g(x)Rp dx. Ω

(5) We denote H 1 (Ω, Rp ) the Hilbert space of square integrable Rp -valued functions with square integrable ﬁrst partial (generalized) derivatives. The dot product is deﬁned for any f, g ∈ H 1 (Ω, Rp ) by f, g

H1

f, g2 +

k ∂f i=1

∂g , ∂xi ∂xi

. 2

3. Measuring distances on the image space. 3.1. Inﬁnitesimal transformations. Let us consider a space JW of functions ¯ and taking values on Rd , which will be explicitly deﬁned later. To deﬁned on Ω, somewhat ﬁx the ideas, we shall speak of elements of JW as “images” and use the

20

´ AND LAURENT YOUNES ALAIN TROUVE

corresponding photometric vocabulary, although our constructions apply to generic graphs of vector-valued functions. We want to build a distance, denoted hereafter dJW , on JW through a Riemannian analysis. Let j ∈ JW and h ∈ R, and consider a small perturbation jh of j such that jh (x) = j(x − hv(x)) + hσ 2 z(x) + o(h), where v is a displacement ﬁeld and z is an Rd -valued function on Ω. Here and in the following, σ 2 is a ﬁxed positive parameter. The transformation from j to jh is therefore divided in two complementary processes. The ﬁrst, which we call the “geometric transformation,” is a pure deformation of the support for which a point located at x in the ﬁrst image is pushed to location x + hv(x). The second process, called the “photometric transformation,” is the residual, obtained by the addition of σ 2 hz. Both transformations are the main ingredients of any morphing process between two images. When j is smooth, we have (1)

∂j jh − j = σ 2 z − dj(v). lim ∂h |h=0 h→0 h

∂j is an element of the tangent The usual geometric interpretation is that γ ∂h |h=0 space Tj JW , and, given our representation, it is sensible to let the length |γ|j depend on w (z, v) and to let w vary in some chosen vector space W . The solution cannot merely be to set |γ|j = |w|W , where | |W is a norm on W , because the representation (z, v) → γ is not one-to-one: if w = (v , z ) is such that

σ 2 (z − z) − dj(v − v) = 0,

(2)

then the transformations along w and w of j are inﬁnitesimally equivalent. Hence, looking for the best tradeoﬀ between geometric and photometric transformations, we can choose for the metric on the tangent space Tj JW (3) |γ|j = inf |w|W | w = (v, z), γ = σ 2 z − dj(v) . Now, we can deﬁne formally (4)

dJW (j0 , j1 ) inf 0

1

∂j

, j path from j0 to j1 .

∂t

jt

3.2. Diﬀerentiable structure. The previous construction is now made rigorous for JW L2 (Ω, Rd ). Remark 1. Since L2 (Ω, Rd ) is a Hilbert space, it has a natural structure of smooth inﬁnite dimensional manifold. However, the diﬀerential structure we need to consider here is diﬀerent from the standard L2 structure. To see this, consider the following example: Ω =]0, 1[k , and jh (x) j0 (x − hv(x)), where • j0 (x) 1x1 ≥1/2 , • v ∈ Cc∞ (Ω, Rk ) is such that the ﬁrst coordinate, v1 , of v is strictly positive at the center c (1/2, . . . , 1/2) of Ω. Then, |jh − j0 |2 /h → +∞ so that jh is not diﬀerentiable at h = 0 for the usual L2 diﬀerentiable structure, whereas, by the construction above, it will be so for the differential structure on JW (this is a justiﬁcation for keeping the nonstandard notation JW for the image space). Our construction starts with the deﬁnition of C 1 paths on JW . We ﬁrst need to specify the allowed geometric as well as grey-level inﬁnitesimal transformations.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

21

3.2.1. Inﬁnitesimal transformation spaces. Geometric transformation. We denote B the space of the displacement ﬁelds underlying the inﬁnitesimal geometric transformation. We assume that B is a Hilbert space with dot product denoted by , B and norm denoted by | |B . We assume throughout this paper that B is continuously embedded in C0p (Ω, Rk ), where p = 1 at least but may be larger if speciﬁed. As a reminder, we recall that B is continuously embedded in some Banach space B (with norm | |B ) of functions if and only if each element v of B can be considered as an element of B and there exists a constant C such that, for all v ∈ B, |v|B ≤ C |v|B . Moreover, B is compactly embedded in B if it is continuously embedded and any bounded set for the norm on B is relatively compact in the B -topology. We shall also assume that Cc∞ (Ω, Rk ) is dense in B. Photometric transformation. Grey-level transformations are assumed to belong to the space L2 (Ω, Rd ). Finally, we denote W B × L2 (Ω, Rd ) on which we place the dot product deﬁned for w = (v, z) and w = (v , z ) by w, w W v, v B + σ 2 z, z 2 . 3.2.2. Diﬀerentiable curves and tangent space. For any smooth image j, we have, for any u ∈ Cc∞ (Ω, Rd ) and any w = (v, z) ∈ W , σ 2 z − dj(v), u2 = σ 2 z, u2 + j, div(u ⊗ v)2 ,

(5)

where div(u⊗v) ∈ C0 (Ω, Rd ) is deﬁned by div(u⊗v)i = div(ui v). The right-hand side of the equality is well deﬁned for arbitrary j ∈ JW , which leads us to the following deﬁnition. Definition 2 (C 1 curves in JW ). Let I be an interval in R. We say that j : I → JW is a continuously diﬀerentiable curve if there exists w (v, z) ∈ C(I, W ) such that (1) j ∈ C(I, L2 (Ω, Rd )) for the usual L2 -topology, (2) for any u ∈ Cc∞ (Ω, Rd ), t → jt , u2 is a continuously diﬀerentiable real-valued ∂ function and ∂t jt , u2 = σ 2 zt , u2 + jt , div(u ⊗ vt )2 . If we deﬁne as usual tangent vectors via classes of ﬁrst order equivalent curves, we can identify the tangent bundle of JW from the deﬁnition of C 1 path on JW as follows. Definition 3. (1) For any j ∈ JW and any u ∈ Cc∞ (Ω, Rd ), we denote lj,u the continuous linear form on W (the continuity stems from the continuous embedding of B in C01 (Ω, Rk )) deﬁned for any w = (v, z) ∈ W by (6)

lj,u (w) σ 2 z, u2 + j, div(u ⊗ v)2 .

(2) We deﬁne (7)

Ej { w ∈ W | lj,u (w) = 0, ∀u ∈ Cc∞ (Ω, Rd ) },

and (8)

Tj JW {j} × W/Ej ,

where W/Ej is the quotient space, the elements of which are denoted w.

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´ AND LAURENT YOUNES ALAIN TROUVE

Remark 2. The use of a quotient space is a consequence of the nonuniqueness of the representation of the derivative by an element w ∈ W as explained by (2). We consider Tj JW as a vector space where for any γ = (j, w) and γ = (j , w ) ∈ Tj JW , we have γ + λ γ (j, w + λw ). Now, if we deﬁne T JW

T j JW ,

j∈JW

T JW plays the role of the tangent bundle of the manifold JW . Definition 4. (1) We denote π : T JW → JW the canonical projection deﬁned by π(γ) = j for any γ (j, w) ∈ Tj JW . (2) Let γ (j, w) ∈ T JW and w = (z, v) ∈ w. For any u ∈ Cc∞ (Ω, Rd ), we denote γ , u σ 2 z, u2 + j, div(u ⊗ v)2 . (Note that the right-hand side does not depend on the choice of w ∈ w). (3) For any function γ : I → T JW where I is a real interval, we say that γ is measurable if π ◦ γ is measurable from I to JW and for any u ∈ Cc∞ (Ω, Rd ), γt , u is measurable from I to R. Returning to Deﬁnition 2, we see that C 1 curves j admit a lifting t → γt = (jt , wt ) to T JW such that for all u ∈ Cc∞ (Ω, Rd ) d jt , u2 = γt , u dt so that it is natural to deﬁne (9)

djt dt

γt ∈ Tjt JW leading to the formula

d jt , u2 = dt

djt ,u . dt

The next step, for our Riemannian construction, is to place a metric on Tj JW for all j ∈ JW . 3.3. Riemannian structure. Definition 5. For any j ∈ JW , we deﬁne on Tj JW the norm |γ|j inf{ |w|W | (j, w) ∈ γ }. The inﬁmum is attained at a unique point, as stated in the following proposition. Proposition 1. For any j ∈ JW and any γ = (j, w) ∈ Tj JW , since w is a closed subspace of W , there exists a unique w ∈ W denoted p(γ) such that p(γ) Argmin |w|W . w∈w

Hence, |γ|j |p(γ)|W . Moreover, p is linear from Tj JW to W . Proof. Since w is a close subspace of W , it is suﬃcient to note that if p is the orthogonal projection from W to Ej⊥ , then p(w) = 0 for any w ∈ Ej so that p can be factorized as a linear map p from W/Ej to Ej⊥ . Now, one easily checks that p(γ) ∈ w and that p(γ) minimizes the norm.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

23

We can now deﬁne the geodesic distance between arbitrary points j0 , j1 in JW by

1

dj 1

dt | j ∈ Cpw dJW (j0 , j1 ) inf (10) ([0, 1], JW ), j0 = j0 , j1 = j1 ,

dt

0

jt

1 where Cpw ([0, 1], JW ) is the set of piecewise C 1 curves in JW which are straightforwardly deﬁned from the deﬁnition of C 1 curves. This deﬁnition is the usual deﬁnition for ﬁnite dimensional Riemannian manifolds. There is, however, a measurability issue, since it is not obvious from our deﬁnition of a measurable path in T JW that t → |γt |π(γt ) is measurable. This issue is addressed in Proposition 2, the proof of which is provided in Appendix A. Proposition 2. Let γ : [0, 1] → T JW be a measurable path in T JW . Then, p ◦ γ is a measurable path in W and |γ|π◦γ is a measurable real-valued function.

4. Groups of diﬀeomorphisms. Curves in W naturally generate diﬀeomorphisms on Ω by integration of their ﬁrst component, which is a time-dependent vector ﬁeld on Ω which vanishes at ∂Ω. The relations between the Hilbert structure on B and the class of diﬀeomorphisms which can be generated in that way have been investigated, in particular, in [30] and [16], in which suﬃcient smoothness conditions on the vector ﬁeld are derived to ensure existence, uniqueness, and smoothness of the ﬂow for all time. For T > 0, deﬁne the set L1 ([0, T ], B) as the Banach space of measurable functions v : [0, T ] → B such that |v|1,T

T

|v|B dt < ∞.

0

Similarly, L2 ([0, T ], B) denotes the Hilbert space of square integrable functions deﬁned on [0, T ] and taking values in B, with the norm

|v|2,T

0

T

1/2 2 |v|B

dt

.

For v ∈ L1 ([0, T ], B), consider the ODE (11)

dy = vt (y). dt

A global ﬂow solution of this equation is a time-dependent family of functions t → ϕt such that, for all x ∈ Ω, ϕ0 (x) = x and

t

vs ◦ ϕs ds.

ϕt = 0

When the dependence of this ﬂow on v must be emphasized, it is denoted by ϕv . Results in [30, 16] essentially relate the existence and smoothness of such ﬂows to embedding conditions of B into standard sets of continuous functions. We quote these results in the following theorem. Theorem 1 (Trouv´e). If B is continuously embedded in C01 (Ω, Rk ), then for all T > 0 and all v ∈ L1 ([0, T ], B), the ODE (11) can be integrated over [0, T ], and its associated ﬂow ϕv is such that at all times x → ϕvt is a homeomorphism of Ω.

24

´ AND LAURENT YOUNES ALAIN TROUVE

Notation 1. Assume that B is continuously embedded in C01 (Ω, Rk ), and introduce the map AT : L1 ([0, T ], B) → C(Ω, Rk ), v → ϕvT . Then, the set A1 (L1 ([0, 1], B)) will be denoted GB . The fact that GB is a group is proved in [30]. Further results on these groups and on AT can be found in Appendix C. The relation between algebraic and metric properties of groups of diﬀeomorphisms and some of the fundamental equations of ﬂuid mechanics has been the subject of several studies, starting with [5], in which the Euler equation is related to the geodesic equations of groups of diﬀeomorphisms with an L2 metric on its Lie algebra (see also [3, 4, 24]). Another important equation, the Camassa–Holm equation, which describes the motion of the waves in shallow water, can be interpreted along the same lines with an Hα1 metric on the Lie algebra [11, 17]. Here, since the energy derives from both geometric and photometric variations, the geodesic equations that we derive can be formally interpreted as conservation of momentum on a semidirect product of the group of diﬀeomorphisms and the space of images, as studied in [25]. However, our point of view of smooth deformations acting on nonsmooth images requires a speciﬁc approach. This is also related to developments in optimal design [28]. 5. Geodesics on JW . 5.1. Minimizing geodesics. The space of C 1 curves is not well suited to deal with proofs of the existence of curves of minimal length between two images j0 and j1 , i.e., minimizing geodesics. We introduce below the more tractable space of curves with square integrable speed. We need ﬁrst a preliminary proposition saying that square integrable paths in T JW are uniquely identiﬁed by their trace on smooth space-time vector ﬁelds in Rd . The proof of this proposition is postponed to Appendix A. Proposition 3. Let γ : [0, 1] → T JW be a measurable path in T JW . Then, if 1 1 2 ∞ d |γ t |π(γt ) dt < +∞ and, for any u ∈ Cc (Ω×]0, 1[, R ), we have 0 γt , ut dt = 0, 0 then γ = 0 a.e. We can now introduce the space H 1 ([0, 1], JW ) of regular curves. Definition 6. We say that a path j ∈ C([0, 1], L2 (Ω, Rd )) is regular if there exists 1 a measurable path γ : [0, 1] → T JW such that π(γ) = j, 0 |γt |2 dt < ∞, and, for any 1 1 u ∈ Cc∞ (]0, 1[×Ω, Rd ), we have − 0 jt , ∂u ∂t 2 dt = 0 γt , ut dt. From Proposition 3, ∂j γt , we get the integration by the path γ is uniquely deﬁned; using the notation ∂t parts formula 1 1 ∂j ∂u (12) jt , dt = − , ut dt. ∂t 2 ∂t 0 0 We denote H 1 ([0, 1], JW ) as the set of all the regular paths in C([0, 1], L2 (Ω, Rd )). Proposition 4. We have C 1 ([0, 1], JW ) ⊂ H 1 ([0, 1], JW ) and both deﬁnitions of ∂j ∂t coincide. Proof. Let j ∈ C 1 ([0, 1], JW ). There exists w = (v, z) ∈ C([0, 1], W ) such that for any u ∈ Cc∞ (Ω, Rd ), t → jt , u2 is C 1 and ∂ jt , u2 = σ 2 zt , u2 + jt , div(u ⊗ vt )2 . ∂t

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

25

Certainly, w ∈ L2 ([0, 1], W ). Moreover, for any u ∈ Cc∞ (Ω, Rd ) and any f ∈ Cc∞ (]0, 1[, R), we have by integration by parts (we denote f (t) df dt ) 1 1 1 d jt , f (t)u2 = f (t)jt , u2 dt = − f (t) jt , u2 dt dt 0 0 0 so that (12) is true for u ⊗ f ∈ Cc∞ (]0, 1[×Ω, Rd ). The complete proof follows by usual density arguments. We carry on with an important result which characterizes regular paths in JW . For a path v in L1 ([0, 1], B), we deﬁne for any s, t ∈ [0, 1] ϕvt,s ϕvs ◦ (ϕvt )−1 . Theorem 2. A path j : [0, 1] → JW is regular (resp., is in C 1 ([0, 1], JW )) if and only if there exists w = (v, z) ∈ L2 ([0, 1], W ) (resp., ∈ C([0, 1], W )) such that t jt = j0 ◦ ϕvt,0 + σ 2 zs ◦ ϕvt,s ds. 0

Proof. The proof is postponed to Appendix B. Theorem 3. Let j0 and j1 be in JW . Then we have

1

∂j

1

(13) dJW (j0 , j1 ) = inf

∂t dt | j ∈ H ([0, 1], JW ), j0 = j0 , j1 = j1 . 0 jt Proof. Let j ∈ H 1 ([0, 1], JW ) be a regular path from j0 to j1 and let w ∈ ∂j L ([0, 1], W ) such that wt = pjt ( ∂t ) for any t. There exists a sequence (wn = 1 (vn , zn ) ∈ C([0, 1], W ), n ∈ N) such that 0 |wt − wtn |2W dt → 0. Deﬁne t n n zns ◦ ϕvt,s ds. jnt = j0 ◦ ϕvt,0 + σ 2 2

0

We get from Theorem 2 that jn ∈ C 1 ([0, 1], JW ). Now, considering w ˜ n (˜ vn , ˜zn ) n n n vn n n with ˜zt zt + (j1 − j1 ) ◦ ϕs,1 and v ˜ v we get from Theorem 9 (see Appendix C) that w ˜ n ∈ C([0, 1], W ). Using Theorem 2, we deduce that if ˜jn is deﬁned by t n n ˜zns ◦ ϕvt,s ds, ˜jnt = j0 ◦ ϕvt,0 + σ 2 0

then ˜j ∈ C ([0, 1], JW ) and = j1 . However, 1 n

1 1

∂˜j

n

dt ≤

| w ˜ | dt → |wt |W dt t W

∂t n 0 0 0 ˜j n

1

˜jn1

t

1 ∂j when n → ∞. Therefore, we deduce that dJW (j0 , j1 ) ≤ 0 | ∂t |jt dt for any regular path from j0 to j1 . Finally, since C 1 ([0, 1], JW ) ⊂ H 1 ([0, 1], JW ), we get the result. Definition 7. Let j0 , j1 ∈ JW . We say that j ∈ C([0, 1], L2 (Ω, Rd )) is a minimizing geodesic path from j0 to j1 if j is regular and

12 1

2

∂j dt = dJW (j0 , j1 ).

∂t

0 jt We denote GJW (j0 , j1 ) as the set of the minimizing geodesic paths from j0 to j1 .

´ AND LAURENT YOUNES ALAIN TROUVE

26

5.2. Characterization of geodesics. 5.2.1. Photometric optimality. Theorem 4. Let j0 , j1 ∈ JW and j ∈ GJW (j0 , j1 ) be a minimizing geodesic path from j0 to j1 . Let w = (v, z) ∈ L2 ([0, 1], W ) be deﬁned by wt p( ∂j ∂t ) for any t ∈ [0, 1]. Then z ∈ C([0, 1], L2 (Ω, Rd )) and for any t ∈ [0, 1] we have

zt = z0 ◦ ϕvt,0 dϕvt,0 .

(14)

Proof. Let j ∈ H 1 ([0, 1], JW ) be a minimizing geodesic from j0 to j1 , and let dj w = (v, z) ∈ L2 ([0, 1], W ) such that for any t ∈ [0, 1], wt = p( dt ). For any u ∈ Cc∞ (]0, 1[×Ω, Rd ) and any ε ∈ R, deﬁne t ˜jt = j0 ◦ ϕvt,0 + σ 2

zs + ε 0

∂us ◦ ϕvs,1 ∂s

◦ ϕt,s ds.

v 2 t Since t → (vt , zt + ε ∂u j ∈ ∂t ◦ ϕt,1 ) ∈ L ([0, 1], W ), we get from Theorem 2 that ˜ 1 H ([0, 1], JW ). Moreover, ˜j0 = j0 and ˜j1 = j1 so that

0

1

2

djt

dt =

dt

1

0

jt

2 |vt |B

+σ

2

2 |zt |2

2

d˜jt

dt

dt

˜jt

1

dt ≤ 0

1

≤

2 |vt |B

0

2

∂ut v

◦ ϕt,1 dt. + σ z t + ε ∂t 2

2

Since ε is arbitrary, we get 0= 0

1

∂ut ◦ ϕvt,1 zt , ∂t

zt ◦

dt = 2

1

0

ϕv1,t

v ∂ut

dϕ1,t , ∂t

dt. 2

d 2 d Choosing arbitrary u ∈ Cc∞ (]0, 1[×Ω,

R ), we get that there exists z˜1 ∈ L (Ω, R ) such that t-a.e. we have zt ◦ ϕv1,t dϕv1,t = z˜1 . Hence, if ˜zt = z˜1 ◦ ϕvt,1

dϕvt,1 , we have ˜z ∈ C([0, 1), L2 ([0, 1], Rd )) and zt = ˜zt t-a.e. Note that ˜z0 ◦ ϕv1,0 dϕv1,0 = z˜1 so that

˜zt = ˜z0 ◦ ϕv1,0 dϕv1,0 ◦ ϕvt,1 dϕvt,1 = ˜z0 ◦ ϕvt,0 dϕvt,0 , and the proof is ended. This leads to the following deﬁnition. Definition 8. A regular path j ∈ H 1 ([0, 1], JW ) is called a pregeodesic path if and only if the following equations are satisﬁed almost everywhere in t:

(15)

⎧ t ⎪ v 2 ⎪ jt = j0 ◦ ϕt,0 + σ ⎪ zs ◦ ϕvt,s ds, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨

zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dj ⎪ ⎪ . ⎩ (vt , zt ) = p dt

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

27

5.2.2. Study of the geodesic equation. Directional derivatives in L2 . In this section, we try to clarify the last equation of system (15), at least in some situations of interest. The diﬃculty comes from the fact that, unless jt is smooth enough, this equation does not, in general, specify a unique correspondence zt → vt . To be more precise, let us analyze the condition that, for all t, dj . (vt , zt ) = p dt For this purpose, we ﬁrst introduce a weak version of the directional derivative dj.v when j ∈ JW and v ∈ B. Definition 9. Let j ∈ JW . (1) We deﬁne the operator Dj : Dj → L2 (Ω, Rd ) by Dj { v ∈ B | ∃C, s.t. ∀u ∈ Cc∞ (Ω, Rd ), |j , div(u ⊗ v)2 | ≤ C |u|2 }, and for any v ∈ Dj , Dj.v is the unique element in L2 (Ω, Rd ) such that Dj.v , u2 = −j , div(u ⊗ v)2 Cc∞ (Ω, Rd ).

for any u ∈ (2) We deﬁne the adjoint operator Dj ∗ : D∗j → B, where D∗j { u ∈ L2 (Ω, Rd ) | ∃C, s.t. ∀v ∈ Dj |Dj.v , u2 | ≤ C |v|B }, and, for any u ∈ D∗j , Dj ∗ is the unique element in Dj (closure of Dj ) such that Dj ∗ .u , vB = u , Dj.v2

(16)

for any v ∈ Dj . Remark 3. The existence of Dj.v comes from the extension of the linear form u → j , div(u ⊗ v) for smooth u into a continuous linear form on L2 (Ω, Rd ) for v ∈ Dj . For the deﬁnition of the adjoint Dj ∗ , the adjoint is uniquely deﬁned as an element of Dj by (16) (Dj is not necessarily dense in B). Fix j ∈ JW . We may characterize elements v ∈ Dj as follows. (We denote hereafter ϕv. as the ﬂow associated with the constant speed vt ≡ v for any t ∈ [0, 1].) Theorem 5. The vector ﬁeld v ∈ B belongs to Dj if and only if there exists a square integrable function ξ : Ω → Rd such that t

−1

−1

(17) ξ ◦ ϕv0,s (x) dϕv0,s (x) ϕvs,t ds. j ◦ ϕv0,t (x) = j(x) dx ϕv0,t + 0

We have in such a case Djv = ξ − jdiv(v). Proof. We ﬁrst notice that, if v ∈ B,

d d v v

j(x) , u ◦ ϕε,0 (x) dx ϕε,0 dx = j ◦ ϕv0,ε (x) , u(x) dx. −j , div(u ⊗ v)2 = dε Ω dε Ω Assuming that (17) holds, the last expression yields ε

−1

−1

d d

j(x) , u(x) dx ϕv0,ε dx + ξ ◦ ϕv0,s , u(x) dϕv0,s (x) ϕvs,ε dx dε Ω dε Ω 0 =− j(x) , u(x)div(v)dx + ξ(x) , u(x)dx = ξ − jdiv(v) , u2 , Ω

Ω

´ AND LAURENT YOUNES ALAIN TROUVE

28

which implies that v ∈ Dj and Djv = ξ − jdiv(v). Conversely, let v ∈ Dj and ξ = Djv+ jdiv(v). Fix u ∈ C 1 (Ω, Rd ). Consider the function f , deﬁned on [0, 1] by f (t) = j ◦ ϕv0,t , u 2 . Denote by ˜j(t) the left-hand term of (17), and g(t) = ˜jt , u2 . We have

−1

g (t) = − j , udivϕv0,t (x) v dx ϕv0,t

+ ξ ◦ ϕv0,t , u 2 2 t

−1

ξ ◦ ϕv0,s dϕv0,s ϕvs,t , udivϕv0,t v ds − 0 2 v = ξ ◦ ϕ0,t − ˜jt divϕv0,t v , u . 2

Since f (t + ε) = j ◦ ϕvt,t+ε , u ◦ ϕvt,0 dϕvt,0 2 , we have

f (t) = Djv , u ◦ ϕvt,0 dϕvt,0 2 = (Djv) ◦ ϕv0,t , u 2 . Therefore, computing the integral of the diﬀerence and using the deﬁnition of ξ,

j ◦ ϕv0,t − ˜jt , u

t

2

= 0

j ◦ ϕv0,s − ˜js , udivϕv0,t v

2

ds ≤ |u|2 |v|B

0

t

j ◦ ϕv0,s − ˜js ds. 2

Taking the supremum of the left-hand term over continuously diﬀerentiable u with L2 -norm equal to 1 yields

j ◦ ϕv0,t − ˜jt ≤ |v| B 2

t

0

j ◦ ϕv0,s − ˜js ds, 2

which implies j ◦ ϕv0,t − ˜jt 2 = 0 for all t. An interesting consequence of this is the following lemma. Lemma 1. For any j ∈ L2 (Ω, Rd ), one has j ∈ Dj∗ and, for v ∈ Dj , Djv , j2 = −

1 2 |j| , divv . 2 2

Proof. Indeed, let v ∈ Dj . Consider the function

j ◦ ϕv0,t (x) 2 dx. f (t) = Ω

2 Since f (t) = |j|2 , |dϕt,0 | 2 , we have f (0) = −|j| , divv2 . Using, on the other hand, expression (17) yields f (0) = 2Djv , j2 . Interpretation of the pregeodesic equations. The property that w = (v, z) ∈ W belongs to Ej , which states that, for all u ∈ Cc∞ (Ω, Rd ), σ 2 z, u2 + j, div(u ⊗ v)2 = 0 is equivalent to v ∈ Dj and σ 2 z − Dj.v = 0. Consider now some tangent vector γ ∈ Tj JW , and study the property that, for some w = (v, z) ∈ W , one has p(γ) = w. This implies, in particular, that, for all (v , z ) ∈ Ej , |v + v |B + σ 2 |z + z |2 ≥ |v|B + σ 2 |z|2 , 2

2

2

2

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

29

which is in turn equivalent to the following: for all v ∈ Dj , v , v B +z , Dj.v 2 = 0. This implies that z ∈ D∗j and that v , v B +Dj ∗ z , v B = 0, so that v = −Dj ∗ z+γ⊥ , where γ⊥ is the projection of v onto D⊥ j . Note that this orthogonal component does not depend on the choice of (v, z) from the equivalence class deﬁning γ (hence the notation). We thus may conclude that (v, z) = p(γ) if and only if z ∈ D∗j and v = −Dj ∗ z + γ⊥ . The ﬁrst conclusion we may draw from this is that, whenever Dj is dense in B, dj vt is uniquely determined by zt and the condition (vt , zt ) = p( dt ). It is given by vt = −Dj∗t zt . This is true, for example, when jt ∈ H 1 (Ω, Rd ) at all times, since, in this case Djt = B (notice that, by Theorem 4, this is true along a geodesic as soon as j0 and j1 belong to H 1 (Ω, Rd )). However, this is not the general situation. As an example, consider the case when j is the indicator function of a subdomain Ω1 of Ω with smooth boundary. If v is a vector ﬁeld on Ω and u is a smooth function on Ω, we have j , divuv = uv , ν1 Rk dσ1 , ∂Ω1

where ν1 is the outward normal to ∂Ω1 and σ1 is the surface measure on ∂Ω1 . This implies that djv may be identiﬁed to a singular measure supported to ∂Ω1 , which does not belong to L2 unless it vanishes. Thus, Dj consists exactly of vector ﬁelds on Ω which belong to B and have vanishing normal components on ∂Ω1 . For a ∈ Rk and x ∈ R, denote by Kx a the element of B such that Kx a , uB = u(x) , aRk . Then, Kx ν(x) belongs to Dj⊥ for any x ∈ ∂Ω, and so does any linear combination of these vector ﬁelds. We see that in this case Dj⊥ is nontrivial. This discussion implies that the pregeodesic condition for a path may be written ⎧ t ⎪ v 2 ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, ⎪ t 0 t,0 ⎪ ⎪ 0 ⎪ ⎨

⎪ zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt ∈ Dj∗ , and vt − Dj∗t zt ∈ Dj⊥t .

(18)

These equations are not complete yet (in the sense that they cannot be solved from the initial values (j0 , v0 , z0 )) since they provide no information on the choice of vt − Dj∗t zt at time t (unless of course Dj⊥t = {0}). We need to specify the mode of propagation of this singular component along a geodesic. The following computation provides a hint on possible ways to achieve this. Assume that j is pregeodesic and dj (vt , zt ) = p( dt ). In such a case, we have 2

|zs |2 =

|z0 |2 |dϕv0,s |−1 dy

|z0 ◦ ϕvs,0 |2 |dϕvs,0 |2 dx = Ω

Ω

and z0 (x) =

1 (j1 ◦ ϕv0,1 (x) − j0 (x)) σ2

0

1

|dϕv0,s |−1 ds.

´ AND LAURENT YOUNES ALAIN TROUVE

30 Thus

1

0

2 |zs |2

1 ds = 4 σ

2

j1 ◦ ϕv0,1 − j0

. 1 |dϕv0,s |−1 ds Ω 0

Making the change of variables y = ϕv0,1 (x) yields 0

1

2 |zs |2

j1 − j0 ◦ ϕv1,0 2

1 ds = 4 σ

Ω

1 0

|dϕv1,0 (x) ϕv0,s |−1 |dϕv1,0 (x)|−1 ds

,

i.e.,

1

0

2 |zs |2

1 ds = 4 σ

2

j1 − j0 ◦ ϕv1,0

, 1 |dϕv1s |−1 ds Ω 0

and the geodesic energy is given by (19) 0

1

2 |vs |B

1 ds + 2 σ

2

j1 − j0 ◦ ϕv1,0

. 1 |dϕv1,s |−1 ds Ω 0

We can obtain more precise information on the geodesic by studying variations of this expression with respect to v. This will be handled below, under a smoothness assumption on j0 . Before this, we need some notation for the reproducing kernel on B. They will be useful throughout the paper. Kernels for the inner-product on B. Proposition 5. There exists a continuous operator K (resp., K∇ ) on L1 (Ω, Rk ) (resp., L1 (Ω, R)) with values in B such that, for all u ∈ L1 (Ω, Rk ) (resp., u ∈ L1 (Ω, R)), for all v ∈ B, Ku , vB = u, v2 , and K∇ u , vB = −u, divv2 . Proof of Proposition 5. Let u ∈ L1 (Ω, Rk ). Since we assume that B is continuously embedded in C01 , the linear form deﬁned on B by v → u, v2 is continuous because |u, v2 | ≤ |u|1 |v|∞ . Therefore, there exists a unique element in B, denoted Ku, such that, for all v ∈ B, Ku , vB = u, v2 and continuity comes from the inequality Ku , vB ≤ |u|1 |v|∞ ≤ cst |u|1 |v|B . The same proof holds for K∇ , since |divv|∞ is also controlled by |v|B . It can be remarked that, for smooth u, K∇ u = K(∇u). Remark 4. When j is smooth (e.g., j ∈ H 1 (Ω, Rd )), the operator Dj ∗ introduced in the previous paragraph is given by Dj ∗ z = K(dj ∗ .z), in which dj ∗ is the standard matrix adjoint of dj. Indeed, we have in this case z , Dj.v2 = z , dj.v2 = dj ∗ .z , v2 = K(dj ∗ .z) , vB .

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

31

Characterization with a smooth endpoint. We study the eﬀect of small variations in v on the geodesic energy (19), under the additional hypothesis that j0 ∈ H 1 (Ω, Rd ). Thus, ﬁx h ∈ L2 ([0, 1], B), and consider a perturbation v + εh of v. We compute the corresponding variation of the geodesic energy. The variation of the 1 ﬁrst term being 2 0 vt , ht B dt, we can focus on the second term, namely, 1 U 2 σ ε

2

j0 ◦ ϕv+εh − j1

1,0 dx. 1 v+εh −1 |dϕ | ds Ω 1s 0

The variations of U ε are given in Lemma 2, which is proved in Appendix E. Lemma 2. We have, at ε = 0, (20) 1 dU ε 2 2 2 =σ K∇ (qtv |zt | ) + K(|zt | ∇qtv ) + 2K([dϕvt,0 ]∗ dϕvt,0 j0∗ zt ), ht dt, dε B 0 t with qtv = 0 |dϕvt,s |−1 ds. We can deduce from this our additional condition for a regular path to be a minimizing geodesic: for almost all t ∈ [0, 1], vt +

1 (KDtv + K∇ Ctv )B = 0, 2

where Dtv σ 2 |zt | ∇qtv + 2[dϕvt,0 ]∗ dϕvt,0 j0∗ zt , 2

and 2

Ctv σ 2 qtv |zt | . It may be interesting to check that this condition boils down to the one we have obtained before for smooth trajectories, namely, vt + K(djt∗ zt ) = 0. It suﬃces to notice that, for pregeodesic trajectories, jt = j0 ◦ ϕvt,0 + σ 2 zt qtv and that, when zt is smooth, KDtv + K∇ Ctv = K(Dtv + ∇Ctv ). We now deﬁne geodesic paths (not necessarily minimizing). Definition 10. Let j0 ∈ H 1 (Ω, Rd ). A regular path j ∈ H 1 ([0, 1], JW ) starting at j0 is called a geodesic path if and only if there exists w = (v, z) ∈ L2 ([0, 1], W ) such that the following equations are satisﬁed almost everywhere in t: t ⎧ v 2 ⎪ ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, t 0 ⎪ t,0 ⎪ ⎪ 0 ⎪ ⎪ ⎨

(21) zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎩ v + K([dϕv ]∗ d v j ∗ z ) + σ K (q v |z |2 ) + K(|z |2 ∇q v ) = 0, t ϕt,0 0 t ∇ t t t t,0 t 2

32

´ AND LAURENT YOUNES ALAIN TROUVE

t with qtv = 0 |dϕvt,s |−1 ds. These equations are complete; it will be shown in section 7 that initial conditions (j0 , z0 ) uniquely specify the solutions. It is interesting to check that geodesics as deﬁned in (21) also are pregeodesics. For this, we ﬁrst show that, for all times t, zt ∈ D∗jt . Noting that the ﬁrst equation in (21) may also be written jt = j0 ◦ ϕvt,0 + σ 2 zt qt it is clear that Djt = Dzt , and zt ∈ Dz∗t is proved in Lemma 1. The same lemma also provides the fact that, for w ∈ Dzt , σ2 2 2 |zt | , div(qtv w) , zt , Djt w2 = zt , d(j0 ◦ ϕvt,0 )w 2 + σ 2 |zt | ∇qt , w − 2 2 and this is equal to −vt , wB by deﬁnition of K and K∇ . We thus obtain the fact that vt + Djt∗ zt ∈ Dj⊥t as required. We shall prove existence of solutions for a broader class of evolution equations, extending the range of initial values v0 . Consider the term ut = K([dϕvt,0 ]∗ dϕvt,0 j0∗ z0 ◦

ϕvt,0 dϕvt,0 ) which appears in the third equation of (21). We have, letting ω0 = −dj0∗ z0 , and, for w ∈ B,

ut , wB = [dϕvt,0 ]∗ dϕvt,0 j0 z0 ◦ ϕvt,0 dϕvt,0 , w 2 L

v

v v

= dϕvt,0 j0 z0 ◦ ϕt,0 dϕt,0 , dϕt,0 w L2 v −1 v = ω0 , (dϕ0,t ) w ◦ ϕ0,t L2 . We know, by Appendix C, that ϕv0,t belongs to C p (Ω) as soon as B is continuously embedded in C0p (Ω, Rk ), which implies in this case (with p ≥ 1) that

(dϕv0,t )−1 w ◦ ϕv0,t

p−1,∞

≤ Const |w|B ,

the constant depending on |v|1,B . But this implies in turn that, if the L2 inner product is replaced by the action of any continuous functional, ω0 , on C0p−1 (Ω, Rk ), which will be denoted

ω0 , (dϕv0,t )−1 w ◦ ϕv0,t ,

there exists an element of B that we shall still denote ut such that ut , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t . With this notation, we may formulate the following deﬁnition. Definition 11. Let j0 ∈ L2 (Ω, Rd ). Let ω0 be a continuous linear functional on p−1 (Ω, Rk ) and z0 ∈ L2 (Ω, Rd ). A regular path j ∈ H 1 ([0, 1], JW ) starting at j0 with C initial direction (ω0 , z0 ) is called a generalized geodesic if and only if, for all u ∈ Dj0 , one has (ω0 , u) + z0 , Dj0 u = 0,

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

33

and there exists w = (v, z) ∈ L2 ([0, 1], W ) such that the following equations are satisﬁed almost everywhere in t: ⎧ t ⎪ v 2 ⎪ ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, t 0 ⎪ t,0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ zt = z0 ◦ ϕv dϕv , ⎨ t,0 t,0 (22) ⎪ ⎪ ⎪ σ2 2 2 ⎪ v v v ⎪ v K − u + (q |z | ) + K(|z | ∇q ) = 0, t ∇ t t ⎪ t t t ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∀w ∈ B uv , w = ω , (dϕv )−1 w ◦ ϕv 0 t 0,t 0,t B t with qtv = 0 |dϕvt,s |−1 ds. Recall that when j0 is smooth, the only choice is ω0 = dj0∗ z0 , and if z0 is also smooth, the system may be written under the simple form ⎧ t ⎪ v 2 ⎪ jt = j0 ◦ ϕt,0 + σ zs ◦ ϕvt,s ds, ⎪ ⎪ ⎪ 0 ⎪ ⎨

(23) zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vt + σ 2 K(djt∗ zt ) = 0. As an example of the nonsmooth applications we have in mind, assume that j0 is a binary, plane image, which is the indicator function of the interior of a connected open subset Ω1 of Ω with smooth boundary ∂Ω1 . We have seen that any element w ∈ Dj0 should be tangent to ∂Ω1 and that in this case Dj0 w = 0 and D∗j0 = L2 (Ω, R). We therefore may choose z0 arbitrarily in L2 , and (ω0 , w) should vanish for w ∈ Dj0 , which is true, for example, when ω0 is deﬁned by (ω0 , w) = w , ν1 dσ1 , ∂Ω1

where ν1 is the outward normal to ∂Ω1 and σ1 is the surface measure on ∂Ω1 . 6. Existence of minimizing geodesics. The next theorem states that minimizing geodesics always exist between two elements of JW . Theorem 6. Assume that B is compactly embedded in C01 (Ω, Rd ), and let j0 , j1 ∈ JW . Then GJW (j0 , j1 ) is nonempty. Proof. Let (jn )n∈N be a minimizing family of paths in H 1 ([0, 1], JW ) from j0 to j1 ; n for any n ∈ N, let wtn p( djdt ) so that (wn )n∈N is a bounded sequence in L2 ([0, 1], W ). Up to the extraction of a subsequence, we can assume that wn converges weakly to a w∞ in L2 ([0, 1], W ). By lower semicontinuity, we have 1 |wt∞ |2W dt ≤ dJW (j0 , j1 ). 0

By a time change argument, which is classical in the proof that minimizing geodesics travel at constant speed (see [12]), we may furthermore assume that |wtn |W is uniformly bounded by, say, dJW (j0 , j1 ) + 1. Denoting wn = (vn , zn ), consider 2 t ∞ ∞ jt j0 ◦ ϕ∞ z ◦ ϕ∞ is the ﬂow associated to v∞ . Since j t,s ds, where ϕ t,0 + σ 0 s

´ AND LAURENT YOUNES ALAIN TROUVE

34

is a regular path, it is suﬃcient to prove that j1 = j1 . However, if ϕn denotes the ﬂow associated with vn , we know, from Theorem 9, that ϕn1,0 converges uniformly n ∞ 2 d ∞ d to ϕ∞ 1,0 so that j0 ◦ ϕt,0 → j0 ◦ ϕt,0 in L (Ω, R ). Now, let u ∈ Cc (Ω, R ). We

1 n 1 have 0 zs ◦ ϕn1,s , u2 ds = 0 zsn , u ◦ ϕns,1 dϕns,1 2 ds. Since u has bounded derivatives and using Theorem 9 implies the uniform convergence of ϕns1 to ϕ∞ s1 and the pointwise convergence of the derivatives (because of the uniform boundedness of |vsn |B ), we have

1

zns , u

(24) 0

◦

ϕns,1

n

dϕs,1 2 ds −

1

∞

zns , u ◦ ϕ∞ s,1 dϕs,1 2 ds → 0.

0

Moreover, from the weak convergence of zn to z∞ , we get

1

(25) 0

∞

zns , u ◦ ϕ∞ s,1 dϕs,1 2 ds →

1

∞

∞

z∞ s , u ◦ ϕs,1 dϕs,1 2 ds,

0

so that ﬁnally j1 − j1 , u2 = 0 for any u ∈ Cc∞ (Ω, Rd ). Hence j ∈ H 1 ([0, 1], JW ) and the result is proved. 7. Initial value problem for the geodesic equation. We have the following theorem. Theorem 7. Assume that B is continuously embedded in C0p (Ω, Rp ) for p ≥ 3. Then, for all T > 0, there exists a unique solution (v, j, z) of (21) over [0, T ], with initial values j0 ∈ H 1 (Ω, Rd ), z0 ∈ L2 (Ω, Rd ), and ω0 ∈ C p−1 ([0, 1], Rk ) (where C p−1 ([0, 1], Rk ) denotes the topological dual of C p−1 ([0, 1], Rk ) with the norm |ω| sup|v|p−1,∞ ≤1 (ω, v)) which continuously depends on these initial conditions. Continuity of the solution (v, j, z) as a function of (j0 , z0 ) is meant according to H 1 (Ω, Rd ) × L2 (Ω, Rd )-norms for the initial conditions, L2 ([0, T ], W )-norm for (v, z), and C([0, 1], L2 (Ω, Rd ))-norm for j. 8. Proof of Theorem 7. To prove Theorem 7, we show the existence of solutions for short time and then extend them to all time. Fix T > 0. For a given v ∈ L2 ([0, T ], B), let Ψ(v) ∈ L2 ([0, T ], B) be deﬁned by

(26)

⎧ σ2 2 2 ⎪ ⎪ Ψ(v)t = uvt − K∇ (qtv |zvt | ) + K(|zvt | ∇qtv ) , ⎪ ⎪ 2 ⎪ ⎨

v v

v

dϕ , = z ◦ ϕ z 0 ⎪ t t,0 t,0 ⎪ ⎪ ⎪ ⎪ ⎩ v ut , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t .

To estimate the Lipschitz coeﬃcient of Ψ, we introduce v, v ∈ L1 ([0, T ], B) and compute the variation of each term in Ψ(v)t − Ψ(v )t . Fix w ∈ B with |w|B = 1. We have σ2 v 2 v σ2 v 2 |zt | , qt div(w) − |zt | , dqtv w + ω0 , (dϕv0,t )−1 w ◦ ϕv0,t 2 2 2 2 2

σ −1 2 |z0 | , dϕv0,t qtv ◦ ϕv0,t div(w) ◦ ϕv0,t = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t + 2 2 σ 2 2

v

−1 v v |z0 | , dϕ0,t − dϕv0,t qt w ◦ ϕ0,t . 2 2

Ψ(v)t , wB = (27)

35

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

We have qtv

◦

ϕv0,t (x)

t

t

−1

v

v v

v

=

dϕ0,t (x) ϕt,s ds =

dϕ0,s (x) ϕs,t ds = 0

v and letting ξs,t =

0

dx ϕv0,t

dx ϕv ds, 0,s

|dx ϕv0,t | and λvt (w) = (dϕv0,t )−1 w ◦ ϕv0,t , |dx ϕv0,s |

(28) Ψ(v)t , wB =

0

t

σ2 2

t

−1

−1 v

2 |z0 | , ( dϕv0,s div(w) ◦ ϕv0,t − dϕv0,t

∇ξs,t , λvt (w) ) ds

0

+(ω0 , λvt (w)). This implies σ2 2 |Ψ(v )t − Ψ(v)t |B ≤ |z0 |2 sup 2

σ2 2 |z0 |2 sup + 2

t 0

t

v −1

dϕ0,s div(w) ◦ ϕv0,t

v −1 v − dϕ0,s div(w) ◦ ϕ0,t ds : |w|B = 1

v −1 v

dϕ0,t

∇ξs,t , λvt (w)

0

v −1 v v − dϕ0,t

∇ξs,t , λt (w) ds : |w|B = 1

+ |ω0 | sup λvt (w) − λvt (w)

: |w|B = 1 . p−1,∞

The problem is thus reduced

−1 to the estimation of variations, with respect to v, of v λvt (w), ∇ξs,t and of dϕv0,s div(w) ◦ ϕv0,t . They involve diﬀerentials of ϕv , ϕv , and w up to the second degree. The inclusion of B in C 3 ([0, 1], Rk ) and an application of Lemmas 7 and 11 in the appendix directly lead to the estimate C max(|v|1,T ,|v | ) 2 1,T , (29) |Ψ(v)t − Ψ(v )t |B ≤ C σ 2 |z0 |2 + |ω0 | |v − v |1,T e and ﬁnally √ C max(|v|1,T ,|v | ) 2 1,T |Ψ(v) − Ψ(v )|2,T ≤ C T σ 2 |z0 |2 + |ω0 | |v − v |1,T e √ C T max(|v|2,T ,|v | ) 2 2,T . (30) ≤ CT σ 2 |z0 |2 + |ω0 | |v − v |2,T e Therefore, Ψ is q-Lipschitz with q < 1 for T small enough, and its unique ﬁxed point yields a unique solution of (21). This is stated below. Lemma 3. There exists a time T > 0 depending only on |z0 |2 and |j0 |H 1 such that a unique solution of (21) exists on [0, T ]. We now show that this solution can be extended to all times. For this, we prove that there exists a unique ﬁxed point for Ψ at all times. Denote by ΨT this mapping when deﬁned on L2 ([0, T ], B). Clearly, if v is a ﬁxed point of ΨT , its restriction to [0, S] is a ﬁxed point of ΨS . Thus, if T0 is the largest T such that ΨS has a unique ﬁxed point vS in L2 ([0, S], B) for any S < T , then each vT for T < T0 is an extension of vS whenever S ≤ T . We can show that T0 = ∞ by showing that, if T0 < ∞, then

´ AND LAURENT YOUNES ALAIN TROUVE

36

there exists ε > 0 (depending only on T0 and the initial conditions) such that, for all T < T0 , there exists a unique extension of vT to [T, T + ε]. Fix such a T ; the issue of extending a ﬁxed point of ΨT on [T, T + ε] can be rephrased as a ﬁxed point problem for small time with the following notation. For v ∈ L2 ([0, T ], B) and v ∈ L2 ([0, ε], B), deﬁne v ∨ v ∈ L2 ([0, T + ε], B), equal to v on [0, T ] and equal to (t → v (t − T )) on [T, T + ε]. Introduce the function Ψε : L2 ([0, ε], B) → L2 ([0, ε], B) deﬁned by Ψε (v)(t) = ΨT +ε (vT ∨ v)(t − T ). For t > T , qtv

T

∨v

T

v = q1v + qt−T ,

zt = zT ◦ ϕvT t |dϕvT t |−1 , and

uvt , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t = ω0 , (dϕv0,T )−1 (dϕv0,T ϕvT t )−1 w ◦ ϕvT t ◦ ϕv0,T = ωT , dϕvT t )−1 w ◦ ϕvT t with (ωT , w) = ω0 , (dϕv0,T )−1 w ◦ ϕv0,T . It is clear that the study of Ψε can follow exactly the lines of the study of ΨT , yielding a unique ﬁxed point if ε is small enough, the size of admissible ε being controlled by the L2 -norms of zT and the norm of ωT as a linear form on C0p−1 (Ω, Rk ). These norms can in turn be bounded by the L2 norms of z0 and the norm of ω0 , respectively, multiplied by a continuous function T vT |1,∞ , |ϕvT,0 |1,∞ ). Proving that this is uniformly bounded for T < T0 is of max(|ϕ0,T therefore suﬃcient to get the contradiction we aim for, that is, that the solution can be uniquely extended beyond T0 . T T So, everything relies on proving the uniform boundedness of ϕv0,T , ϕvT,0 , and their derivatives over Ω. By Lemmas 7 and 9, these quantities are bounded by functions of |vT |1,T so that we have to prove that these can be bounded uniformly in T . However, it suﬃces to use the facts that vT satisﬁes a geodesic equation and that geodesics travel at constant speed. More precisely, deﬁning, for t ≤ T < T0 ,

2 2 ψt = vtT B + σ 2 |zt |2 , we have (recall that this does not depend on T as soon as T ≥ t) ψt ≡ ψ(0) so that

T

v

≤ T ψ(0) 1,T for all T . It is well known that minimizing geodesics have constant speed, but we must check that this property remains true for all the solutions of (22). This is proved in the appendix and is stated, for further reference, in the next lemma. 2 2 Lemma 4. If (j, v, z) is a solution of system (22) on [0, T [, then |vt |B + σ 2 |zt |2 is constant with respect to time. To prove the continuity of the solution, let (v, j, z) and (v , j , z ) be two solutions of system (22) with initial conditions (ω0 , z0 ) and (ω0 , z0 ), respectively. Using, in particular, the computation leading from (28) to (29), it is not to diﬃcult to obtain the estimate

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | eC|v|1,T 2 C |v | 2 1,T . + C σ 2 |z0 |2 + |ω0 | |v − v |1,T e

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

37

As we just have shown, |v|1,T = T |v0 |B , and this is smaller (up to a universal multiplicative constant) than |ω0 | so that

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | eCT |ω0 | 2 2 2 + C σ |z0 |2 + |ω0 | |v − v |1,T eCT |ω0 | . Gronwall’s lemma now allows us to conclude that, for some constant C which may now depend on T, |ω0 | , |ω0 | , |z0 |2 , and |z0 |2 , (31)

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | . 2

9. Normal coordinates in H 1 . We now consider the question, which motivated this paper, of whether the previous construction could be used as an indexing device for characterizing the deformations and variations of an object relative to a prototype. Fix an image j0 ∈ H 1 (Ω, Rd ). The computationally simplest way to describe an image j in a neighborhood of j0 is by the diﬀerence j − j0 . However, one cannot be satisﬁed with this representation which takes no account of the metric we have placed on JW . Among local charts related to the metric, normal coordinates on a Riemannian manifold are radial ﬂattenings of this manifold onto its tangent space in the sense that radial lines in this space correspond to geodesics on the manifold. They provide a very eﬃcient linear representation of the manifold and of its metric. Existence of such coordinates is a standard theorem in ﬁnite dimensions, and our purpose is to check how much of this result remains valid in our inﬁnite dimensional framework. In the previous sections, another candidate for local coordinates has emerged, which turns out to be closely related (it is in fact dual) to normal coordinates. We have proved that, for a ﬁxed j0 ∈ H 1 (Ω, Rd ), one can associate to any z0 ∈ L2 (Ω, Rd ) a unique solution of system (21). We introduce the function Mj0 : L2 → L2 , which assigns to z0 ∈ L2 the “image” j1 , where jt is the solution of (21) at time t. The following theorem shows that Mj0 shares some features of local coordinates on JW . Theorem 8. Let BH 1 (0, ε) denote the open ball in H 1 (Ω, Rd ) containing all z0 ∈ H 1 (Ω, Rd ) such that |z0 |H 1 < ε. Then, for all j0 ∈ H 1 (Ω, Rd ), there exists ε > 0 such that Mj0 restricted to BH 1 (0, ε) is continuous and one-to-one onto its image, equipped with the L2 -topology. Proof of Theorem 8. Continuity of Mj0 : L2 (Ω, Rd ) → L2 (Ω, Rd ) is a consequence of Theorem 7, and it trivially implies the continuity of the restriction Mj0 : H 1 (Ω, Rd ) → L2 (Ω, Rd ) for the H 1 (Ω, Rd )-topology. We show that this map is one-to-one in a neighborhood of 0. We ﬁrst have the following lemma. z0 |H 1 ) ≤ 1. Denote v ˜ the Lemma 5. Let j0 , z0 , z˜0 ∈ H 1 (Ω, Rd ), with max(|z0 |H 1 , |˜ time-dependent vector ﬁeld along the solution of (21) with initial condition (j0 , z˜0 ). Then, there exist a constant C and a function ε which depend only on j0 such that, for t > 0,

˜

(Mj (t˜ z0 ) − Mj0 (tz0 )) ◦ ϕv0,t − t[σ 2 (˜ z0 − z0 ) + dj0 K(dj0∗ (˜ z0 − z0 ))] 2 0 ≤ Ct |˜ z0 − z0 |2 ε(t),

´ AND LAURENT YOUNES ALAIN TROUVE

38

and limt→0 ε(t) = 0. The proof of Lemma 5 is given in Appendix G. To prove Theorem 8, we ﬁrst remark that

2

σ (˜ z0 − z0 ) + dj0 K(dj0∗ (˜ z0 − z0 )) 2 ≥ σ 2 |˜ z0 − z0 |2 so that

C v ˜

2

(Mj (t˜ 1 − z ) − M (tz )) ◦ ϕ ≥ tσ |˜ z − z | ε(j , t) , 0 j0 0 0 0 2 0 0,t 2 0 σ2

and the lower bound is nonvanishing as soon as t is small enough. Remark that we have, for j1 , j2 ∈ H 1 (Ω, Rd ), the inequality d(j1 , j2 ) ≤

1 |j1 − j2 |2 σ

since the right-hand side is an upper bound of the length of the curve jt = (1−t)j1 +tj2 2 2 1 t (since choosing v ≡ 0 and σ 2 z ≡ j2 − j1 , we have wt (vt , zt ) ∈ ∂j |zt |2 = ∂t and σ 0 2 |j2 − j1 |2 /σ 2 ). So continuity of Mj0 for the d-topology on its image is also true. According to Lemma 5, normal coordinates (which are time derivatives at t = 0 of geodesics) are related to M by the relation (we use the standard exponential notation) expj0 (Sz0 ) = Mj0 (z0 ), where S is deﬁned by Sz σ 2 z + Dj0 K(Dj0∗ z). This indicates that a good approximation of the metric in terms of the z-coordinate would be 2

|z1 − z2 |j0 = z1 − z2 , S(z1 − z2 )2 , which satisﬁes |z1 |j0 = d(j0 , Mj0 (z1 )) in a neighborhood of 0. 10. Experiments. In this section, we propose a preliminary set of experiments to illustrate the information contained in the z-coordinate described above. Experiments in Figures 1, 2, and 3 were conducted in two steps: given two images j0 and j1 , we ﬁrst computed the minimizing geodesic between them, yielding a trajectory (jt , zt , vt ) and the corresponding ﬂow ϕvt . Then, using j0 again, and the obtained value z0 on the minimizing geodesic, we computed the solution of (21) until time t = 1. The obtained values (jt , zt , vt ) could then be compared with those which have been computed along the geodesics. In Figure 4, the initial j0 is the same as in Figures 2 and 3, but the z0 is the average of the two so that it does not correspond to any precomputed geodesic in the image space. The result is quite interesting, because it still possesses characteristics of a human face and can be compared to the result of a simple linear combination of the target images in Figures 2 and 3. The numerical implementation of both operations (minimization of the geodesic energy and integration of (21)) must be done with some care in order, in particular, to avoid instabilities due to the conservation part of the energy. Details will be provided in a forthcoming paper.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

39

Fig. 1. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

Appendix A. Proofs of Propositions 2 and 3. A.1. Proof of Proposition 2. The proof relies on a sequence of standard measurability arguments, of which we sketch only the main steps. First let (wn )n∈N be a Hilbert basis of W . Since, for any u ∈ Cc∞ (Ω, Rd ) and w = (v, z) ∈ W , j → lj,u (w) (which has been deﬁned in (6) by σ 2 z, u2 + j, div(u ⊗ v)2 ) is continuous from L2 (Ω, Rd ) to R, the map j → wj,u lj,u (wn )wn n≥0

is measurable from L2 (Ω, Rd ) to W . By construction, we have, for w ∈ W , w , wju W = lj,u (wn )w , wn W = lj,u (w). n≥0

Thus, for γ ∈ Tj JW , we have p(γ) = Argmin |w|W : w , wj,u = γ , u for all u ∈ Cc∞ (Ω, Rd ) . Introducing a family (un )n∈N in Cc∞ (Ω, Rd ) which is dense in H01 (Ω, Rd ), the previous expression may be replaced by p(γ) = Argmin {|w|W : w , wj,un = γ , un for all n ≥ 0} .

´ AND LAURENT YOUNES ALAIN TROUVE

40

Fig. 2. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

For N ∈ N and λ > 0, we deﬁne (32)

pN,λ (γ) = Argmin

2 |w|W

+λ

N

w , wj,un W

2 − γ , un

.

n=0

N Clearly, we must have pN,λ (γ) = i=1 xi wj,ui , where ⎧

2

N 2 N N ⎨

x = Argmin

xn wj,un + λ xn wj,un , wj,un W − γ , ui

x ∈RN +1 ⎩

n=0

n=0

W

n =1

⎫ ⎬

+

1 2 |x | . ⎭ λ

For λ > 0, the optimal x is given by x = (A + I/λ)−1 y, where y ∈ RN +1 is such that yi = γ , ui and A is an (N + 1) × (N + 1) matrix with coeﬃcients given

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

41

Fig. 3. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

by an,n = wj,un , wj,un W . This implies that, if γt is a measurable path, the function t → pN,λ (γt ) is measurable. The measurability of p(γt ) is a consequence of the pointwise convergence of pN,N

(γt ) to p(γt ), which comes from the following argument: for all N and λ, we have pN,λ (γ) W ≤ |p(γ)|W , since the last term in (32) vanishes for w = p(γ). For the same reason, N n=0

2 1 pN,λ (γ) , wj,un − γ , un ≤ |p(γ)|W , λ

which implies that for all n, pN,N (γ) , wj,un → γ , un when N tends to inﬁnity. Moreover, for any weakly converging subsequence extracted from pN,N (γ) (which forms a weakly set in W ), we have, and denoting w∗ its limit,

N,N compact

∗

|w |W ≤ lim inf p (γ) W ≤ |p(γ)|W , and, for all n, w∗ , wj,un = γ , un by weak convergence, which is only possible when w∗ = p(γ). Hence t → p(γt ) is measurable if γt is measurable, and the proof of Proposition 2 is ended.

´ AND LAURENT YOUNES ALAIN TROUVE

42

Fig. 4. From top to bottom and from left to right: Initial image, target image, z-coordinate, obtained by averaging the z-coordinate in Figures 2 and 3, and obtained target image.

A.2. Proof of Proposition 3. We deduce from Proposition 2 that it is suﬃcient to prove the next proposition. Proposition 6. Let w ∈ L2 ([0, 1], W ) such that for any u ∈ Cc∞ (Ω×]0, 1[, Rd ) we have (33)

1

σ 2 zt , ut 2 + jt , div(ut ⊗ vt )2

dt = 0.

0

Then almost everywhere in t, wt ∈ Ejt . Proof. Let (un )n∈N be a family in Cc∞ (Ω, Rd ) dense in Cc∞ (Ω, Rd ) for the H 1 (Ω, Rd )norm. If we prove that for any n ∈ N, the function cn deﬁned by cn (t) σ 2 zt , un 2 + jt , div(un ⊗ vt )2 is vanishing almost everywhere, then by density, there exists a negligible set N such that for any t ∈ [0, 1] \ N and any u ∈ Cc∞ (Ω, Rd ) σ 2 zt , u2 + jt , div(u ⊗ vt )2 = 0,

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

43

which implies Proposition 6. Hence, let us consider n ∈ N. For any f ∈ Cc∞ ([0, 1], R), if u(t, x) f (t)un (x), we have from (33) that 1 cn (t)f (t)dt = 0 0

so that, by standard arguments, we get cn = 0 almost everywhere. Appendix B. Proof of Theorem 2. We start the (⇐) part in the case L2 ([0, 1], W ). Lemma 6. Let w = (z, v) ∈ L2 ([0, 1], W ). Let us deﬁne for any t ∈ [0, 1] t 2 jt j0 ◦ ϕt,0 + σ zs ◦ ϕt,s ds, 0

where ϕt is the ﬂow at time t associated with v. Then j is regular. Proof. Let us notice ﬁrst that t+h jt+h = jt ◦ ϕt+h,t + σ 2 (34) zs ◦ ϕt+h,s dt. t

From equality (34), the continuity in JW of j is straightforward. It is suﬃcient to prove that for any u ∈ Cc∞ (Ω×]0, 1[, Rd ), we have 1 1 2 ∂u jt , − dt = (35) σ zt , ut 2 + jt , div(ut ⊗ vt )2 dt. ∂t 2 0 0 Indeed, if (35) is proved, if for any t ∈ [0, 1] we denote γt (jt , wt ), we have for any u ∈ Cc∞ (Ω, Rd ), t → γt , u = σ 2 zt , u2 + jt , div(u ⊗ vt )2 measurable, and 1 1 |γt |jt ≤ |wt |W so that 0 |γt |2t dt ≤ 0 |wt |2W dt < +∞, and the lemma is proved. We have 1 1 1 jt+h − jt ∂u ut − ut−h , ut dt jt , jt , dt = − lim − dt = lim h→0 0 h→0 0 ∂t 2 h h 0 2 2 so that 1 t+h 1 1 ∂u 2 − jt ◦ ϕt+h,t − jt , ut 2 dt + σ jt , = lim zs ◦ ϕt+h,s , ut 2 ds dt. h→0 h 0 ∂t 2 0 t t+h However, jt ◦ ϕt+h,t − jt , ut 2 = t jt ◦ ϕs,t , div(ut ⊗ vs )2 ds so that 1 t+h 1 1 ∂u 2 jt , dt = lim jt ◦ ϕs,t , div(ut ⊗ vs )2 + σ zs ◦ ϕt+h,s , ut 2 ds dt − h→0 0 h ∂t 2 0 t

t+h 1 1 = lim jt , div(ut ⊗ vs ) ◦ ϕt,s |dϕt,s |2 + σ 2 zs , ut ◦ ϕs,t+h |dϕs,t+h |2 ds dt. h→0 0 h t Since jt is uniformly bounded on L2 and |ϕt,s − I|1,∞ = (|t − s|) (since B is continuously embedded in C 1 (Ω, Rk )), there exists C > 0 such that (36)

1

1 1 t+h

jt , div(ut ⊗ vs ) ◦ (ϕt,s |dϕt,s | − I)2 dsdt ≤ C(h) |vt |1,∞ dt

0 h t

0 1 1/2 2 ≤ C (h) (37) |wt |W dt . 0

´ AND LAURENT YOUNES ALAIN TROUVE

44

Now, using again the fact that jt is uniformly bounded in L2 and fact that C([0, 1], L2 (Ω, Rk )) is dense in L2 ([0, 1], L2 (Ω, Rk )), we get

1 1 t+h

jt , div(ut ⊗ vs ) − div(ut ⊗ vt )2 dsdt

(38)

0 h t 1 t+h 1 ≤C |div(ut ⊗ vs ) − div(ut ⊗ vt )|2 dsdt 0 h t → 0 when h → 0. At this point we have proved that 1 1 1 lim (39) jt ◦ ϕt+h,t − jt , ut 2 dt = jt , div(ut ⊗ vt )2 dt. h→0 h 0 0 Still using the fact that |ϕt,s − I|1,∞ = (|t − s|) and the fact that |ut |1,∞ is uniformly bounded, we have 1 t+h 1 1 |zs |2 ds (40) σ 2 zs , ut ◦ (ϕs,t+h |dϕs,t+h | − I)2 dsdt ≤ Cσ 2 (h) 0 h t 0 1 1/2 ≤ Cσ(h) (41) |wt |22 . 0

Finally, since |ut |∞ is uniformly bounded, we get

1 t+h

1 1 t+h

zs − zt , ut 2 dsdt ≤ lim C |zs − zt |2 dsdt = 0. lim

h→0 h→0 0 h t 0 t Hence the proof of the lemma is ended. Let us consider the (⇒) part of Theorem 2 for H 1 ([0, 1], JW ). Let j ∈ H 1 ([0, 1], JW ) be a regular path, and let wt = p( ∂j ∂t ) for any t ∈ [0, 1]. We get from Proposition 2 that w ∈ L2 ([0, 1], W ). Hence, let us deﬁne the new path j by t jt = j0 ◦ ϕt,0 + σ 2 zs ◦ ϕt,s ds, 0

where ϕ is the ﬂow associated with v. From the (⇐) part, we get that j is regular ∂j ∞ d ∞ and that ∂j ∂t = ∂t . Now let u0 ∈ Cc (Ω, R ). For any f ∈ Cc (]0, 1[, R) if u(t, x) = u0 (x)f (t) for any x ∈ Ω and t ∈ [0, 1], we have from the integration by parts formula for a regular path 1 r(t)f (t)dt = 0, 0

jt , u2 .

Since r is continuous and r(0) = 0, we get r ≡ 0. where r(t) = jt , u2 − Considering arbitrary u0 , we get ﬁnally jt = jt for any t ∈ [0, 1]. Since the (⇒) part for C 1 ([0, 1], JW ) is a straightforward consequence of the deﬁnition of C 1 curves and of the (⇒) part for H 1 ([0, 1], JW ), we consider the (⇐) part for w ∈ C([0, 1], W ). We get from the corresponding part for L2 ([0, 1], W ) that (35) is still true. For any f ∈ Cc∞ ([0, 1], R) and any u ∈ Cc∞ (Ω, Rd ) we have 1 t f (t)jt , u2 dt = f (t) σ 2 zt , ut 2 + jt , div(u ⊗ vt )2 dt. − 0

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

45

One easily checks that t → σ 2 zt , ut 2 + jt , div(u ⊗ vt )2 is continuous as well as t → jt , u2 so that, considering smooth approximates of step functions, we deduce that s 2 js , u2 = j0 , u2 + σ zt , ut 2 + jt , div(u ⊗ vt )2 dt, 0

and the result is proved. Appendix C. Regularity results for AT . In this section, we collect a few useful results on how the regularity of the diﬀeomorphism AT (v) = ϕvT may be related to the norm on B, provided this norm can in turn control a suﬃcient number of derivatives; the ﬁrst result deals with boundedness. In the following, we assume at least that B is continuously embedded in C01 (Ω, Rk ) so that AT is well deﬁned for all T . In this case, ϕvT (x)

T

vs (ϕvs (x))ds.

=x+ 0

If vs had p space derivatives for all s, a formal diﬀerentiation of this equality yields (42)

d

p

ϕvT

T

dp (vs ◦ ϕvs )ds.

p

= d id + 0

This can be proved rigorously from rather standard arguments in the study of ODEs and is stated in the next lemma, for which we provide a proof for completeness, because of the small complication due to the fact that we have only an L1 control with respect to the t variable, instead of the usual uniform one. Lemma 7. If p ≥ 1 and B is embedded in C0p (Ω, Rk ), then, for all v ∈ L1 ([0, T ], Ω), v ϕ is p times diﬀerentiable and, for all q ≤ p, ∂ q v d ϕt = dq (vt ◦ ϕvt ). ∂t Moreover, there exist constants C, C such that, for all v ∈ L1 ([0, T ], Ω), (43)

sup |ϕvs |p,∞ ≤ CeC

|v|1,T

.

s∈[0,T ]

Proof. For further reference, we ﬁrst state Gronwall’s lemma. Lemma 8 (Gronwall). Asume that α and β are two positive, continuous functions on the interval [0, c] and that w(t) ≤ α(t) +

t

β(s)w(s)ds. 0

Then, w(t) ≤ α(t) + 0

t

t β(u)du α(s)β(s)e s ds.

´ AND LAURENT YOUNES ALAIN TROUVE

46

The continuity of x → ϕv0,t (x) is a direct consequence of this lemma since, for x, y ∈ Ω,

t

v v v v

vs (ϕ0,s (x)) − vs (ϕ0,s (y)) ds

|ϕ0,t (x) − ϕ0,t (y)| = x − y + 0

t

vs 1,∞ |ϕv0,s (x) − ϕv0,s (y)|ds,

≤ |x − y| + 0

and Gronwall’s lemma implies |ϕv0,t (x) − ϕv0,t (y)| ≤ |x − y| exp(C |v|1,T ).

(44)

Assume p = 1 and pass now to the diﬀerential of ϕv0,t . Fix x ∈ Ω and introduce the linear diﬀerential equation, formally obtained in (42) for p = 1, ∂Wt = dϕv0,t (x) vt Wt ∂t

(45)

with initial condition W (0) = δ ∈ Rk . We skip the argument ensuring the existence and uniqueness of a solution of this equation on [0, 1] and proceed to identifying it as Wt = dx ϕv0,t δ. Denote aε (t) = ϕv0,t (x + εδ) − ϕv0,t (x) /ε − Wt . For α > 0, introduce μt (α) = max {|dx vt − dy vt | : x, y ∈ Ω, |x − y| ≤ α} . The function dx vt ∈ C01 (Ω) being uniformly continuous on the compact set Ω, we have limα→0 μt (α) = 0. We may write t 1 t aε (t) = vs (ϕv0,s (x + εδ)) − vs (ϕv0,s (x)) ds − dϕv0,s (x) vs Ws ds ε 0 0 t dϕv0,s (x) vs aε (s)ds = 0 1 t (vs (ϕv0,s (x + εδ)) − vs (ϕv0,s (x)) − εdϕv0,s (x) vs (ϕv0,s (x + εδ) − ϕv0,s (x)))ds. + ε 0 Since for all y, y ∈ Ω |vt (y ) − vt (y) − dy vt (y − y)| ≤ μs (|y − y|) |y − y| , we may write |aε (t)| ≤ 0

t

|vs |1,∞ |aε (s)| ds + C(v) |δ|

1

μs (εC(v) |δ|)ds 0

for some constant C(v) which depends only on v. The fact that aε (t) tends to 0 when ε → 0 now is a direct consequence of Gronwall’s lemma and of the fact that 1 lim μs (α)ds = 0, α→0

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

47

which is true by the dominated convergence theorem, since μs pointwise converges to 0 and μs (α) ≤ 2 |v|1,∞ . This proves Lemma 7 in the case p = 1. The rest of the proof is by induction: let q0 ≤ p, q0 > 1, and assume that the result is proved for all q < q0 : ∂ q v d ϕt = dq (vt ◦ ϕvt ). ∂t This implies that for δ1 , . . . , δq ∈ Rk , we may write ∂ q v (l) (l) dl vt (δ1 , . . . , δl ), d ϕt (δ1 , . . . , δq ) = dϕv0,t vt dq ϕvt (δ1 , . . . , δq ) + ∂t q

l=2

(l)

each vector δk being a linear combination (with universal coeﬃcients) of terms of the kind dl ϕv0,t (δi1 , . . . , δil ) with l ≤ q + 1 − l (this result on the diﬀerentials of the composition of two functions can be easily proved by induction). This is a linear equation in dq ϕvt (δ1 , . . . , δq ), which is valid for q = q0 − 1, and the proof of its validity for q0 follows exactly the same lines as for p = 1. This expression also shows (using Gronwall’s lemma) that |dq ϕvt |∞ may be bounded by an expression of the kind

|v|1,T C˜ |dϕvt |∞ , . . . , dq−1 ϕvt ∞ exp C |v|1,T , where C˜ is a polynomial, which in turn implies (43). The same estimate is true for (ϕv )−1 . Lemma 9. If p ≥ 1 and B is continuously embedded in C0p (Ω, Rk ), there exist constants C, C such that, for all v ∈ L1 ([0, T ], Ω),

sup (ϕvs )−1 p,∞ ≤ CeC |v|1,T . s∈[0,T ]

Lemma 9 is a consequence of Lemma 7 and of the fact that (ϕvt )−1 = ϕw t with ws = −vt−s on [0, t]. We now pass to suﬃcient conditions for Lipschitz continuity of AT . For this, let ξ v, v ∈ L1 ([0, T ], B). For ξ ∈ [0, 1], denote vξ = (1 − ξ)v + ξv and ϕξ = ϕv . Lemma 10. t ξ ξ ∂ vξ (46) dϕvξ (x) ϕvut (vu − vu ) ◦ ϕvs,u (x)du. ϕs,t (x) = s,u ∂ξ s Proof. Let us ﬁrst start with a formal diﬀerentiation of ξ

∂ϕvs,t ξ = vtξ ◦ ϕvs,t ∂t with respect to ξ, which yields ξ ∂ ∂ vξ d ξ ϕ = (vt − vt ) ◦ ϕvt + dϕvξ vtξ ϕvs,t , s,t ∂t ∂ξ s,t dξ

which naturally leads us to introduce the solution of the diﬀerential equation (47)

ξ ∂ Wt = (vt − vt ) ◦ ϕvt + dϕvξ vtξ Wt s,t ∂t

´ AND LAURENT YOUNES ALAIN TROUVE

48

with initial condition Ws = 0. Noting that we have already encountered this equation ξ without the constant term in (45), the solution of which is of the form dx ϕv0,t δ, a standard argument by variation of the constant shows that the solution of (47) is given by the right-hand term of (46). Therefore, the proof boils down to show that the interversion of derivatives underlying the formal argument above can be made rigorous. For this, it clearly suﬃces to consider the problem in the vicinity of ξ = 0. The proof in fact follows the same lines as the proof of Lemma 7: letting Wt be the solution of (47), we let ξ aξ (t) = ϕvs,t (x) − ϕvs,t (x) /ξ − Wt and express it under the form, letting hu = vu − vu , s

1 + ξ

t

aξ (t) =

t s

t

ξ

(hu (ϕvs,u (x)) − hu (ϕvs,u (x)))du

dϕvs,u vu aξ (u)du +

s

ξ vu (ϕvs,u (x)) − vu (ϕvs,u (x)) − ξdϕvs,u (x) vu ϕvs,u (x) − ϕvs,u (x) du. ξ

The proof can proceed exactly as that of Lemma 7, provided it has been shown that ξ |ϕvs,u (x)−ϕvs,u (x)| tends to 0 with ξ, which is again a direct consequence of Gronwall’s lemma and of the inequality t

ξ

ξ

t

v

|vu |1,∞ ϕvs,u (x) − ϕvs,u (x) du + ξ |hu |∞ du.

ϕs,t (x) − ϕvs,t (x) ≤ s

s

This lemma implies, in particular, that (48)

1

ϕvs,t (x) − ϕvs,t (x) = 0

s

t

d ϕv ξ

s,u (x)

ϕvut (vu − vu ) ◦ ϕvs,u (x)dudξ, ξ

ξ

which almost immediately leads to the following result (by computing diﬀerentials and applying Lemma 7). Lemma 11. Assume that B is continuously embedded in C0p (Ω). If v, v ∈ 1 L ([0, T ], B), we have, for t ≤ T ,

v

ϕt − ϕvt

Cp (|v|1,t +|v |

p−1,∞

≤ Cp |v − v |1,t e

1,t

)

for some constant Cp which depends only on p. √ The same results apply on L2 ([0, 1], B), since |v|1,T ≤ T |v|2,T , but, in this space, weak continuity is true under more general conditions. Theorem 9 (Trouv´e, Dupuis, et al). Assume that B is continuously embedded in C0p (Ω, Rk ). Then the map ˜ T : L2 ([0, T ], B) → C p ([0, T ] × Ω, Rk ), A v → ϕv. (.) is continuous for the weak topology on L2 ([0, 1], B) and the norm |.|T,p−1,∞ on C p ([0, T ]× Ω, Rk ) deﬁned by |ϕ|T,p−1,∞ = ess.sup(|ϕt |p−1,∞ , t ∈ [0, T ]).

49

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Moreover, assume that the embedding is compact, that vn converges weakly to v, and that there exists a constant A such that, for all n and almost all s ∈ [0, 1], |vsn |B ≤ A. Then, for all x ∈ Ω and t ∈ [0, T ], n

dpx ϕvt → dpx ϕvt . Recall that vn converges to v in the weak topology on L2 ([0, 1], B) if and only if, for all w ∈ L2 ([0, 1], B),

1

lim

n→∞

0

Ω

vn (t) , w(t)B dt =

0

1

Ω

v(t) , w(t)B dt.

Proof. The proof of this theorem, which is sketched here for completeness, relies on the remark that, since vn weakly converges, it is bounded in L2 ([0, 1], B), and Lemma 7 readily implies that (ϕvn ) and their space derivatives up to order p − 1 are equicontinuous sequences in space. Equicontinuity in time comes by applying the Cauchy–Schwarz inequality to t dq ϕvt − dq ϕvs = dq (vu ◦ ϕvu )du. s

Ascoli’s theorem implies compactness of (ϕvn ) for the |.|T,p−1,∞ -topology, and it remains to identify a limit of any converging subsequence as ϕv . Denoting this limit by ψ, one deduces from t n vn ϕ0,t (x) = vsn (ϕv0,s (x))ds 0 n

v to ϕt the fact that and the convergence of ϕ0,t

t

vsn (ψs (x))ds + o(n),

ψt (x) = 0

t and the conclusion comes after the remark that w → 0 ws (ψs (x))ds is a continuous linear functional on L2 ([0, 1], B) so that the weak convergence of vn to v implies that ψt (x) =

t

vs (ψs (x))ds 0

and ψt = ϕv0,t . We now prove the pointwise convergence of the pth derivative. We know that d p v d ϕ = dϕvt vdpx ϕvt + Qvt (x), dt x t where Qvt (x) depends on the derivatives of v evaluated at ϕvt (x) and on the p − 1 ﬁrst space derivatives of ϕvt . We may therefore write t t n n dϕvs n v(dpx ϕvs − dpx ϕvs )ds + (dϕvs v − dϕvs n vn )dpx ϕvs ds dpx ϕvt − dpx ϕvt = 0

0

t

n

(Qvs (x) − Qvs (x))ds.

+ 0

´ AND LAURENT YOUNES ALAIN TROUVE

50

t

n

The ﬁrst integral may be bounded by C(|vn |1,T ) 0 dpx ϕvs − dpx ϕvs ds, and the result will be a consequence of Gronwall’s lemma, provided we show that the remaining terms tend to 0. Consider the second integral, which may be written t t (dϕvs v − dϕvs vn )dpx ϕvs ds + (dϕvs vn − dϕvs n vn )dpx ϕvs ds. 0

0

The ﬁrst term tends to 0 because w → 0

t

dϕvs (x) wdpx ϕvs ds

is a continuous linear functional on L2 ([0, t], B) and vn weakly converges to v in this space. To estimate the second one, introduce, for A, ε > 0, the number C(A, ε) = max {|dx w − dy w| : x, , y ∈ Ω, |x − y| ≤ ε, |w|B ≤ A} . The compact embedding assumption implies that, A being ﬁxed, C(A, ε) tends to 0 when ε tends to 0. Using this notation, we have t t n n p v (dϕvs v − dϕvs n v )dx ϕs ds ≤ C (|vsn |B , |ϕvs − ϕvs n |∞ ) |dpx ϕvs |∞ ds 0 0 t C (A, |ϕvs − ϕvs n |∞ ) |v|B ds, ≤ 0

where A = ess.sup {|vsn |B , n ≥ 0, s ∈ [0, 1]}. The last upper bound now tends to 0, by dominated convergence. Finally, a generic term of Qvt being dkϕvt (x) vt (di1 ϕvt , . . . , dik ϕvt ), we can use the same argument to prove its pointwise convergence. Appendix D. Action of diﬀeomorphisms on images. The next theorem provides results concerning the regularity of the action of diﬀeomorphisms on L2 (Ω, Rd ) and H 1 (Ω, Rd ). Theorem 10. (i) Let ϕ be a diﬀeomorphism of Ω such that ϕ and ϕ−1 have uniformly bounded ﬁrst derivatives on Ω. Then, if i ∈ L2 (Ω, Rd ) (resp., i ∈ H 1 (Ω, Rd )), also i ◦ ϕ ∈ L2 (Ω, Rd ) (resp., i ◦ ϕ ∈ H 1 (Ω, Rd ) and dx (i ◦ ϕ) = dϕ(x) i.dx ϕ). (ii) Moreover, for all M > 0 and for all i ∈ L2 (Ω, Rd ), there exists a function εM (i, η) such that, for all ϕ, ϕ such that

−1

−1

max |ϕ|1,∞ , ϕ 1,∞ , |ϕ |1,∞ , ϕ

≤ M, 1,∞

we have |i ◦ ϕ − i ◦ ϕ|2 ≤ εM (i, |ϕ − ϕ |1,∞ ), and εM (i, η) → 0 when η → 0. The same statement is true for i ∈ H 1 (Ω, Rd ), the L2 (Ω, Rd )-norm being replaced by the H 1 (Ω, Rd )-norm.

51

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Proof. We start with (i) and give the proof for H 1 (Ω, Rd ), since it contains exactly the arguments which are valid for L2 (Ω, Rd ). Fix ϕ and let Lϕ be deﬁned by Lϕ (i) = i ◦ ϕ. The vector space Cd∞ = C ∞ (Ω, Rd ) of restrictions to Ω of inﬁnitely diﬀerentiable functions on Rk taking values in Rd is dense in H 1 (Ω, Rd ) [10]. The linear map Lϕ is continuous from Cd∞ (with the topology induced by H 1 (Ω, Rd )) to H 1 (Ω, Rd ); indeed, for i ∈ Cd∞ ,

2 2 |Lϕ (i)|H 1 = |i ◦ ϕ|22 + |dϕ i dϕ|22 ≤ |i|22 dϕ−1 ∞ + |di|22 |dϕ|2∞ |dϕ−1 |∞ ≤ C |i|H 1 since the ﬁrst derivatives of ϕ and ϕ−1 are bounded. Thus, Lϕ restricted to Cd∞ ˜ ϕ on H 1 (Ω, Rd ). If i ∈ H 1 (Ω, Rd ) and in can be extended to a continuous function L ∞ is a sequence of elements of Cd which converges to i when n tends to inﬁnity (so ˜ ϕ (i) in H 1 (Ω, Rd )), then, because convergence in H 1 (Ω, Rd ) implies that in ◦ ϕ → L convergence in L2 (Ω, Rd ), a subsequence of in can be extracted which converges almost ˜ n (i). If N ⊂ Ω everywhere to i and such that in ◦ ϕ converges almost everywhere to L −1 null

Lebesgue measure, then it is also the case for ϕ (N ) (by boundedness of

has −1 ˜ ϕ = Lϕ .

dϕ ), so that in ◦ ϕ also converges almost everywhere to i ◦ ϕ, yielding L 1 2 Now, since the map i → di is obviously continuous from H to L , so is i → d(Lϕ (i)). But, since this map coincides with i → dϕ i dϕ on Cd∞ , and this last map is also continuous on H 1 (Ω, Rd ) (by the previous computation), we get equality over all H 1 (Ω, Rd ), again by density of Cd∞ . For (ii), we ﬁrst consider the L2 (Ω, Rd ) case. Let i, ϕ , ϕ, and M be as in the theorem, and ﬁx s ∈ C ∞ (Ω, Rd ); we have |i ◦ ϕ − i ◦ ϕ|2 ≤ |i ◦ ϕ − s ◦ ϕ |2 + |s ◦ ϕ − s ◦ ϕ |2 + |i ◦ ϕ − s ◦ ϕ|2 . First notice that |i ◦ ϕ − s ◦ ϕ |2 = 2

−1

−1

dϕ |i − s|2 dx ≤ C ϕ

Ω

2

1,∞

|i − s|2

for some constant C. For the middle term, we have

|s ◦ ϕ − s ◦ ϕ |2 ≤

1

0

dϕ+t(ϕ −ϕ)s (ϕ − ϕ) dt 2

≤ |ϕ − ϕ|∞

0

1

dϕ+t(ϕ −ϕ)s dt 2

≤ C(M ) |ds|2 |ϕ − ϕ|∞ . We thus get |i ◦ ϕ − i ◦ ϕ|2 ≤ C(M ) (|i − s|2 + |ds|2 |ϕ − ϕ|∞ ) . Letting εM (i, η) C(M )

inf

s∈C ∞ (Ω)

(|i − s|2 + |ds|2 η)

yields the conclusion of the theorem in the L2 (Ω, Rd ) case, the H 1 (Ω, Rd ) case being handled similarly.

´ AND LAURENT YOUNES ALAIN TROUVE

52

Appendix E. Proof of Lemma 2. We must compute the derivative at ε = 0 of 1 U = 2 σ ε

2

j0 ◦ ϕv+εh − j1

1,0 dx. 1 −1 ds |dϕv+εh Ω 1,s | 0

First, we notice the equation σ 2 zt (x) =

(49)

j1 ◦ ϕvt,1 − j0 ◦ ϕvt,0 , 1 |dϕvt,s |−1 ds 0

which implies that (diﬀerentiating at ε = 0) 1 dU ε d d

v+εh

−1 2 2 = −2 z1 , dϕv1,0 j0 ϕv+εh dϕ |z − σ | , ds. 1 1,s dε dε 1,0 dε 0 2 2 Starting with the ﬁrst term and using Lemma 10, we have 1 d dϕv1,0 j0∗ z1 , dϕv1t ϕvt,0 ht ◦ ϕv1t dt z1 , dϕv1,0 j0 ϕv+εh = − 1,0 dε 2 0 2 1 =− dϕvt,0 j0∗ z1 ◦ ϕvt1 |dϕvt1 | , dϕvt,0 ht dt 0

1

=− 0

=−

1

(dϕvt,0 )∗ dϕvt,0 j0∗ zt , ht

2

dt 2

K (dϕvt,0 )∗ dϕvt,0 j0∗ zt , ht dt B

0

because of the identity zt = z1 ◦ ϕvt1 |dϕvt1 |. We now pass to the second term, for which we use the equality # t $

v+εh −1

dϕt,s = exp div(vu + εhu ) ◦ ϕv+εh du , t,u s

which is a consequence of Lemma 7 and standard computations on the derivative of the determinant. This implies that d

v+εh

−1

v

−1 t dϕt,s = dϕt,s div(hu ) ◦ ϕvt,u du dε s u

−1 t

+ dϕvt,s

dϕvt,u div(vu ) dϕvtτ ϕvτ u hτ ◦ ϕvtτ dτ du s t

v −1 t v div(hu ) ◦ ϕt,u du = dϕt,s

s

v −1 t τ

dϕvt,u div(vu )dϕvtτ ϕvτ u hτ ◦ ϕvtτ dudτ. − dϕt,s s

s

We may notice that

−1

∇ |dϕvτ s |

−1 , ξ = |dϕvτ s | s

τ

dϕvτ u (divvu )dϕvτ u (ξ)du

53

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

to identify the last term as

v −1 t

dϕv ϕvτ s ∇ϕv |dϕvτ s |−1 , hτ ◦ ϕvtτ dτ

dϕt,s

tτ tτ s

so that d

v+εh

−1

v

−1 t dϕt,s = dϕt,s div(hu ) ◦ ϕvt,u du dε s t −1 −1 ∇ϕvtτ |dϕvτ s | , hτ ◦ ϕvtτ dτ. |dϕvtτ | − s

Therefore, 1 1 1 d

v+εh

−1 2 2 −1 dϕ1,s |z1 | , |z1 | , |dϕv1s | div(hu ) ◦ ϕv1u duds ds = dε 2 0 0 s 2 1 1 2 −1 −1 − |z1 | , |dϕv1u | ∇ϕv1u |dϕvus | , hu ◦ ϕv1u duds

s

0

1

0 1

1

s 1

= − 0

−1 2 −1

|zu | , |dϕvu1 | dϕvu1 ϕv1s div(hu ) duds 2

−1

|zu | , ∇ |dϕvus |

2

, hu

s

duds. 2

Introducing quv

u

|dϕvus |−1 ds,

0

this may be written 1 1 d

v+εh

−1 2 2 dϕ1,s |z1 | , quv |zu | , div(hu ) du ds = dε 2 0 0 2 1 2 − |zu | , ∇quv , hu du 0

1

=− 0

1

−

2

2

K∇ (quv |zu | ) , hu 2

K(|zu | ∇quv ) , hu

0

Now, deﬁning functions 2

Ctv σ 2 qtv |zt |

(50) and (51)

Dtv σ 2 |zt | ∇qtv + 2[dϕvt,0 ]∗ dϕvt,0 j0∗ zt , 2

Proposition 5 implies dU ε = dε

0

1

ht , K.Dtv + K∇ Ctv B dt,

B

du

B

du.

2

´ AND LAURENT YOUNES ALAIN TROUVE

54

which is the conclusion of Lemma 2. Appendix F. Proof of Lemma 4. We prove that solutions of system (22) 2 2 travel at constant speed and therefore compute the derivative of |vt |B + σ 2 |zt |2 for such a solution. Starting with the second term, we have zt = z0 ◦ ϕvt,0 dϕvt,0 , which implies, after a change of variables,

−1 2 2

|zt |2 = |z0 |2 dϕv0,t dx. Ω

Using the identity

v

dϕs,t = exp

(52)

t

div(vu ) ◦

ϕvs,u du

,

s

we obtain d 2 |zt |2 = − dt

(53)

Ω

−1 2

|z0 |2 dϕv0,t div(vt ) ◦ ϕv0,t dx.

2

To study the variation of |vt |B , we start with the computation of the derivative of vt , w for a ﬁxed w ∈ B. Applying formula (28) for a solution of (22) yields

−1 v

σ 2 1 2

v

−1 vt , wB = |z0 | , ( dϕ0,s ∇ξs,t , λvt (w) ) ds div(w) ◦ ϕv0,t − dϕv0,t

2 0

v with ξs,t have

+ (ω0 , λvt (w))

= dϕv0,t / dϕv0,s and λvt (w) = (dϕv0,t )−1 w ◦ ϕv0,t . From formula (52), we v ξs,t

t

(divvu ) ◦

= exp

ϕv0u du

,

s

which implies that v dξs,t

=

v ξs,t s

so that

t

dϕv0u (divvu )dϕv0u du

σ 2 t 2

v

−1 |z0 | , dϕ0,s div(w) ◦ ϕv0,t ds vt , wB = (ω0 , λvt (w)) + 2 0 2 t t

−1 σ 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) duds − 2 0 s σ 2 t 2

v

−1 v = (ω0 , λt (w)) + |z0 | , dϕ0,s div(w) ◦ ϕv0,t ds 2 0 2 t u

−1 σ 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) dsdu. − 2 0 0

We now compute the time diﬀerential of each term which appears in this expresv d v sion. Denote λt (w) = dt λt (w). We have v

λt (w) =

d (dϕv0,t )−1 w ◦ ϕ0,t = −(dϕv0,t )−1 dϕv0,t vt w ◦ ϕ0,t + (dϕv0,t )−1 dϕv0,t wvt ◦ ϕ0,t . dt

55

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Next, we have

−1 d t 2

v

−1 2

|z0 | , dϕ0,s div(w) ◦ ϕv0,t ds = |z0 | , dϕv0,t div(w) ◦ ϕv0,t dt 0 t

−1 2

|z0 | , dϕv0,s ∇ϕv0,t (div(w))vt ◦ ϕv0,t ds + 0

and d dt

t 0

u

0

t

= 0

−1 2

|z0 | , dϕv0,s dϕv0,t (divvt )dϕv0,t λvt (w) dsdu

t + 0

−1 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) dsdu

0

u

−1 v 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λt (w) dsdu.

Putting everything together, we have σ2

−1 d v 2

|z0 | , dϕv0,t div(w) ◦ ϕv0,t vt , wB = ω0 , λt (w) + dt 2 σ 2 t 2

v

−1 + |z0 | , dϕ0,s dϕv0,t (divw)vt ◦ ϕv0,t dsdu 2 0 σ 2 t 2

v

−1 − |z0 | , dϕ0,s dϕv0,t (divvt )dϕv0,t λvt (w) dsdu 2 0 σ 2 t u 2

v

−1 v |z0 | , dϕ0,s dϕv0u (divvu )dϕv0u λt (w) dsdu. − 2 0 0 v

d A little care must be taken in writing, as we did above, dt (ω0 , λvt (w)) = (ω0 λt (w)), v v v since this requires proving that (λt+ε (w) − λt (w))/ε converges to λt (w) for the (p − 1, ∞)-norm. This is indeed true in our case, because of the fact that w ∈ B allows us to control the uniform norm of its diﬀerentials up to order p, and the diﬀerentials of ϕvt up to the same order are solutions of a linear diﬀerential equation which ensures their uniform continuity. We now use the identity (which is justiﬁed below)

d 2 |vt |B = 2 lim vt+ε − vt , vt B /ε, ε→0 dt

(54)

2

which implies that, to compute the time diﬀerential of |vt |B , it suﬃces to use the obtained expression for the derivative of vt , wB with w = vt and multiply it by v 2. Since λt (vt ) = 0, and because of (53), we see that all terms cancel, yielding 2 2 d 2 dt (|vt |B + σ |zt |2 ) = 0. To show (54), one writes 2

2

2

(|vt |B − |vt |B − vt+ε − vt , vt B )/ε = |vt+ε − vt |B /ε, and the result is obtained by proving that, for w ∈ B, |vt+ε − vt , wB | = O(ε) |w|B , which can be done by a direct estimation of

d dt vt

, wB .

´ AND LAURENT YOUNES ALAIN TROUVE

56

Appendix G. Proof of Lemma 5. It suﬃces to prove this result for smooth z0 , z˜0 , j0 . It is straightforward that Mj0 (tz0 ) = jt , where j is the solution of (21) with initial conditions (j0 , z0 ). Let ˜jt = Mj0 (t˜ z0 ). Introduce also the corresponding (vt , zt ) and (˜ vt , ˜zt ). Introduce the notation η = ˜j − j, ζ = ˜z − z, and α = v ˜ − v. Since we have assumed smooth trajectories, we may write ∂jt = σ 2 zt − djt vt ∂t and ∂zt = −div(zt ⊗ vt ) ∂t and similar equations for the trajectory with initial condition (j0 , z˜0 ). Computing the diﬀerences along both trajectories yields ⎧ ∂ηt ˜t = σ 2 ζt − djt αt , ⎨ ∂t + dηt v (55) ⎩ ∂ζt ˜t ) = −div(zt ⊗ αt ). ∂t + div(ζt ⊗ v Since & v˜ ∂ζt ∂ %

v˜

˜ ˜ dϕ0,t ζt ◦ ϕv0,t + div(ζt ⊗ v = dϕ0,t

˜t ) ◦ ϕv0,t ∂t ∂t

˜

˜

div(zt ⊗ αt ) ◦ ϕv0,t = − dϕv0,t , the second term yields ζs ◦

˜ ϕv0,s

˜ −1

(˜ = dϕv0,s z0 − z 0 ) −

s

v˜

˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕv0,s ,

0

and the ﬁrst one implies ηt ◦

˜ ϕv0,t

=σ

t

ζs ◦

2

˜ ϕv0,s ds

0

−

t ˜ [djs αs ] ◦ ϕv0,s ds.

0

Replacing ζ in the last equation gives (56) ˜ ηt ◦ ϕv0,t = t[σ 2 (˜ z0 (.) − z0 (.)) − dj0 (˜ v0 − v0 )] t s

v˜

2 v ˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕs,u du ◦ ϕv0,s ds −σ −

0 t

0

˜ [djs αs ] ◦ ϕv0,s − dj0 (˜ v0 − v0 ) ds +

0

t

v˜ −1

dϕ0,s − 1 (˜ z0 − z0 )ds 0

so that Lemma 5 reduces to evaluating the L2 -norm of the last three integrals. We shall use the fact that, for a function f ∈ L2 ([0, 1] × Ω, Rd ),

t

t

fs ds ≤ |fs |2 ds.

0

2

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

57

For t ∈ [0, 1], we also have, from (31), with ω0 = dj0∗ z0 and ω0 = dj0∗ z0 (here and in the following, we denote by const any quantity which depends only on j0 , z0 and z˜0 ), |αt |B ≤ const |z0 − z˜0 |2 .

(57)

This implies that

t s

v˜

˜ v ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕ0,s ds

0 0 2 t s

v ˜

v ˜

˜ ϕ ≤

dϕv0,s s,u div(zu ⊗ αu ) ◦ ϕ0,u duds 2 0 0 t s

v˜ −1

div(zu ⊗ αu ) ds =

dϕ0,s 2 0 0 t s |div(zu ⊗ αu )|2 duds ≤ const 0 0 t s |zu |H 1 |αu |B duds. ≤ const 0

0

v

dϕt,0 implies that |zu | 1 ≤ const |z0 | 1 so that The relation zt = z0 ◦ H H

t s

v˜

˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕv0,s ds

≤ const t2 |˜ z0 − z0 |2 .

ϕvt,0

0

0

2

A similar estimate is valid for the last integral in (56), since ||dϕv0,s |−1 −1|∞ ≤ const s. We ﬁnally consider the second integral in this equation. Since s js = j0 ◦ ϕvs,0 + σ 2 z0 ◦ ϕvs,0 |dϕvu0 | ◦ ϕvs,u du, 0

we have, letting γs =

ϕvs,0

◦

˜ ϕv0,s ,

v ˜ v v ˜ ˜ js α ◦ ϕ ˜ ϕ dϕv0,s 0,s = dγs j0 dϕv s,0 α ◦ ϕ0,s + Rs , 0,s

z0 − z0 |2 . We need to estimate and it is easy to check that |Rt |2 ≤ const t |z0 |H 1 |˜ t (58) 0

v v ˜ ˜ ϕ dγs j0 dϕv0,s ds α ◦ ϕ − dj α s 0 0 s,0 0,s t t v v ˜ ˜ ϕ dγs j0 (dϕv0,s = α ◦ ϕ − α ) ds + (dγs j0 α0 − dj0 α0 ) ds. 0 s,0 s 0,s 0

0

Start with the ﬁrst term, for which we must bound, for the L∞ -norm, the diﬀerence v v ˜ ˜ ϕ dϕv0,s s,0 α ◦ ϕ0,s − α0 or, equivalently, ˜ dϕvs,0 αs − α0 ◦ ϕvs,0 .

It is simple to check, from (57) and estimates we have used several times on the ˜ variations of the diﬀeomorphisms, that (dϕvs,0 − I)αs and α0 ◦ ϕvs,0 − α0 are bounded by const s |˜ z0 − z0 |2 . We now proceed to an upper bound for αs − α0 , for which we need to return to the expression obtained in (28), which yields σ2 s 2 −1 vs − v0 , wB = |z0 | , (|dϕv0u | div(w) ◦ ϕv0,s 2 0

−1 v − dϕv0,s ∇ξus , λvs (w)) du + z0 , dj0 (λvs (w) − w)2

´ AND LAURENT YOUNES ALAIN TROUVE

58 so that

˜ −1 v˜ σ 2 s 2

v˜

−1 ˜

αs − α0 , wB = |˜ z0 | , ( dϕ0u div(w) ◦ ϕv0,s − dϕv0,s ∇ξus , λvs˜ (w) ) du 2 0

−1

σ2 s 2 −1 v − |z0 | , (|dϕv0u | div(w) ◦ ϕv0,s − dϕv0,s ∇ξus , λvs (w)) du 2 0 + z˜0 , dj0 (λvs˜ (w) − w) 2 − z0 , dj0 (λvs (w) − w)2 . The diﬀerence of the ﬁrst two integrals takes the form σ 2 s ˜ z˜0 , Qvus (59) (w) − z0 , Qvus (w) du 2 0

−1

−1 v with Qvus (w) = |dϕv0u | div(w) ◦ ϕv0,s − dϕ v0,s ∇ξus , λvs (w). From Lemmas 7

˜ and 11, and from (57), we obtain the fact that Qvus (w) − Qvus (w) ≤ const |˜ z0 − z0 |2 |w|B so that the quantity in (59) is bounded by const s |˜ z0 − z0 |2 . Writing z˜0 , dj0 (λvs˜ (w) − w) 2 − z0 , dj0 (λvs (w) − w)2 = z˜0 − z0 , dj0 (λvs˜ (w) − w) 2 + z0 , dj0 (λvs˜ (w) − λvs (w)) 2

and using λvs˜ (w) − w ∞ ≤ const s (which is deduced from Lemma 7 and a com

putation of the diﬀerential of λvs˜ (w) with respect to s) and λvs˜ (w) − λvs (w) ∞ ≤ const s |˜ z0 − z0 |2 (from Lemma 11 and (57)), we ﬁnally conclude that

v v ˜ ˜ ϕ α ◦ ϕ ≤ const s |˜ z0 − z0 |2 , − α

dϕv0,s 0

s,0 0,s ∞

which implies that the ﬁrst integral in the right-hand term of (58) is bounded by const t2 |˜ z0 − z0 |2 . Consider now the last term of (58), namely, t (dj0 ◦ γs − dj0 ) α0 ds. 0

Since |α0 |∞ ≤ C |˜ z0 − z0 |2 , we must estimate |dγs j0 − dj0 |2 . By Theorem 10, this is a function of the kind

εM (dj0 , |γs − Id|∞ ) = εM (dj0 , ϕv˜ (s) − ϕv (s) ∞ ), ˜ where M depends only on |j0 |H 1 , |z0 |2 , |˜ z0 |2 . Since |ϕv0,s − ϕv0,s |∞ = O(s), we get (with another function ε) t (dj0 ◦ γs − dj0 ) α0 ds ≤ ε(j0 , t)t |˜ z0 − z0 |2 . 0

We need ﬁnally to consider the last line of (56) which can be easily bounded from above by ε(j0 , t)t |˜ z0 − z0 |2 . We now can collect the estimates we have obtained to conclude the proof of Lemma 5. REFERENCES [1] Y. Amit, U. Grenander, and M. Piccioni, Structural image restoration through deformable templates, J. Amer. Statist. Assoc., 86 (1989), pp. 376–387. [2] Y. Amit and P. Piccioni, A non-homogeneous Markov process for the estimation of Gaussian random ﬁelds with nonlinear observations, Ann. Probab., 19 (1991), pp. 1664–1678.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

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[3] I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. [4] V. Arnold and B. Khesin, Topological methods in hydrodynamics, in Annual Review Fluid of Mechanics, 24, Annual Review, Palo Alto, CA, 1992, pp. 145–166. [5] V. I. Arnold, Sur la g´ eometrie diﬀerentielle des groupes de Lie de dimension inﬁnie et ses applications a ` l’hydrodynamique des ﬂuides parfaits, Ann. Inst. Fourier (Grenoble), 1 (1966), pp. 319–361. [6] R. Bajcsy and C. Broit, Matching of deformed images, in Proceedings of the 6th International Conference on Pattern Recognition, Munich, Germany, 1982, pp. 351–353. [7] R. Bajcsy and S. Kovacic, Multiresolution elastic matching, Comp. Vision, Graphics, and Image Proc., 46 (1989), pp. 1–21. [8] F. L. Bookstein, Principal warps: Thin plate splines and the decomposition of deformations, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), pp. 567–585. [9] F. L. Bookstein, Morphometric Tools for Landmark Data; Geometry and Biology, Cambridge University Press, Cambridge, UK, 1991. [10] H. Brezis, Analyse Fonctionnelle, Th´ eorie et Applications, Masson, Paris, 1983. [11] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), pp. 1661–1664. [12] M. P. D. Carmo, Riemannian Geometry, Birkha¨ user, Basel, 1992. [13] E. Christensen, R. D. Rabbitt, and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Trans. Image Process., 1996, pp. 1435–1447. [14] T. Cootes, C. Taylor, D. Cooper, and J. Graham, Active shape models: Their training and application, Comp. Vis. and Image Understanding, 61 (1995), pp. 38–59. [15] C. David and S. W. Zucker, Potentials, valleys, and dynamic global coverings, Int. J. of Comp. Vision, 5 (1990), pp. 219–238. [16] P. Dupuis, U. Grenander, and M. Miller, Variational problems on ﬂows of diﬀeomorphisms for image matching, Quart. Appl. Math., 56 (1998), pp. 587–600. [17] C. Foias, D. D. Holm, and E. S. Titi, The Navier-Stokes-alpha model of ﬂuid turbulence, Phys. D, 152 (2001), pp. 505–519. [18] D. Geman and S. Geman, Stochastic relaxation, gibbs distribution and bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6 (1984), pp. 721–741. [19] U. Grenander, Lectures in Pattern Theory, Appl. Math. Sci. 33, Springer-Verlag, New York, 1981. [20] U. Grenander, General Pattern Theory, Oxford Science Publications, Oxford University Press, New York, 1993. [21] U. Grenander and D. M. Keenan, Towards automated image understanding, J. Appl. Stat., 16 (1989), pp. 207–221. [22] U. Grenander and D. M. Keenan, On the shape of plane images, SIAM J. Appl. Math., 53 (1993), pp. 1072–1094. [23] U. Grenander and M. I. Miller, Representations of knowledge in complex systems (with discussion section), J. Roy. Statist. Soc. Ser. B, 56 (1994), pp. 549–603. [24] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1994. [25] J. E. Marsden, T. Ratiu, and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), pp. 147–177. [26] M. I. Miller, S. C. Joshi, and G. E. Christensen, Large deformation ﬂuid diﬀeomorphisms for landmark and image matching, in Brain Warping, A. Toga, ed., Academic Press, New York, 1999, pp. 115–131. [27] M. I. Miller, and L. Younes, Group action, diﬀeomorphism and matching: A general framework, Int. J. Comp. Vis, 41 (2001), pp. 61–84. ´ [28] F. Murat and J. Simon, Etude de probl` emes d’optimal design, in Optimization Techniques: Modeling and Optimization in the Service of Man, Lecture Notes in Comput. Sci. 41, Springer-Verlag, New York, 1975, pp. 54–62. [29] B. D. Ripley and A. I. Sutherland, Finding spiral structures in images of galaxies, Phil. Trans. Roy. Soc. A, 332 (1990), pp. 477–485. ´, Inﬁnite Dimensional Group Action and Pattern Recognition, Tech. report, DMI, [30] A. Trouve Ecole Normale Sup´erieure, Paris, France, 1995. ´, Diﬀeomorphism groups and pattern matching in image analysis, Int. J. Comp. [31] A. Trouve Vis., 28 (1998), pp. 213–221. [32] L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), pp. 565–586.

SIAM J. MATH. ANAL. Vol. 37, No. 1, pp. 17–59

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES∗ ´ † AND LAURENT YOUNES‡ ALAIN TROUVE Abstract. In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which inﬁnitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield eﬃcient and robust estimations of diﬀeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this inﬁnite dimensional “shape manifold” on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We ﬁnally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template. Key words. inﬁnite dimensional Riemannian manifolds, deformable templates, shape representation and recognition, warping AMS subject classiﬁcations. Primary, 58b10; Secondary, 49J45, 68T10 DOI. 10.1137/S0036141002404838

1. Introduction. The theoretical developments which are addressed in this paper are motivated by the theory of deformable templates, as it emerged from the work of Grenander and his collaborators in the 1980’s [19, 21, 22, 20], to handle image processing problems. This theory has an abstract formulation, in which the purpose is to represent the variability within an object class by the variations in shape, or color, etc., of a single object, submitted to the action of “deformations.” For instance, a model designed to describe a picture of a human face should be able to explain interindividual variations but also variations caused by the change of expression of a given individual, and by the changing of imaging conditions, such as lighting, occultations, etc. The interesting feature in Grenander’s construction is that it assigns a large part, sometimes all, of the variations to a ﬁxed structure, describing the deformation, which is independent of the particular instance of the observed image. This structure most of the time belongs to a group, the group of deformations, which is acting on the set of objects. The speciﬁc choice of the group depends on the application and on the type of visual features which are modeled, like pixelized images [18] and discretized shapes [20, 29, 23]. In such discrete settings, the group action is used to generate variations of the constituting generators of the object (pixels for an image, segments for polygons) and therefore are modeled as ﬁnite dimensional groups, generally products of linear or aﬃne groups. In the simple example of labeled collections of points (landmarks), the deformation may simply correspond to independent translations of each point, but when the question is raised of the similarity of two collections of landmarks, one would like to ﬁgure out the amount of deformation which is required to transform one of them into the other. When evaluating this deformation, it is clear that the lengths of the induced translations should have some impact, but that this is not the ∗ Received by the editors March 29, 2002; accepted for publication (in revised form) June 11, 2004; published electronically August 17, 2005. http://www.siam.org/journals/sima/37-1/40483.html † LAGA (UMR CNRS 7593), Institut Galil´ ee, Universit´e Paris XIII, Av. J-B. Cl´ ement, F-93430 Villetaneuse, France ([email protected]). ‡ CMLA (CNRS, URA 1611), Ecole Normale Sup´ erieure de Cachan, 61 avenue du Pr´esident Wilson, F-94235 Cachan Cedex, France ([email protected]).

17

18

´ AND LAURENT YOUNES ALAIN TROUVE

only factor and often not even the main factor. One would also like to draw conclusions on the smoothness of the deformation, based on the fact that, in the context of large deformations of shapes, a lower similarity must be associated to a collection of translations which point to erratic directions, compared to a more homogeneous displacement. We see, in this case, that a global point of view on the displacements is needed. Spline-based landmark matching [9, 26] speciﬁcally addresses this issue by seeking the smoothest function which interpolates the considered displacements. When dealing with image deformation, the need to pass to the continuum is even more obvious. In this case, deformations, which should provide nonambiguous point displacements, must be diﬀeomorphisms on the image support. This nonambiguity constraint, however, has been relaxed in most of the early attempts to deal with this issue, working preferably with linear spaces of deformations [6, 7, 8, 2, 1, 14], which can be seen as ﬁrst order approximations. Dealing explicitly with true deformations, i.e., diﬀeomorphisms acting on the support of images, was rigorously formalized by Riemannian metric arguments on the groups of diﬀeomorphisms in [32] for onedimensional problems, and in [31] in full generality (see also [30]). Stemming from the simple representation of right invariant metrics on groups of diﬀeomorphisms along a path in this space, i.e., time-dependent deformations, in terms of the Eulerian velocity, this last reference built diﬀeomorphisms as ﬂows associated to ODEs (a construction which was already present in [3]) and transferred the modeling eﬀort to the linear space of velocities, i.e., of vector ﬁelds deﬁned on the image support. Under suitable Banach space structures on these linear spaces, the extension of the ODE solutions for inﬁnite time and the existence of minimizers to general variational problems in this space can be ensured, providing rigorous suﬃcient conditions for the well-posedness of many practical problems in template matching. This analysis rejoined the line of work of Miller and his collaborators on the estimation of large deformation diﬀeomorphisms [13, 26], in which velocity-based models have been introduced, and variational properties studied in [16]. In [27], the interest in considering a lifted group action, on the cross product of the group itself and of the image space, was demonstrated in a wide variety of applications. The ﬁnal metric on the image space was obtained by projecting a right-invariant Riemannian distance designed on the product space. The approach we follow in this paper addresses the same kind of construction as in [27], which focused on the metric aspects, but from a diﬀerent point of view. Our purpose is to start from the inﬁnitesimal analysis of small deformations of images in order to model and measure image variations and deﬁne diﬀerentiable and geodesic curves in the image space. We shall accept conditions which ensure enough smoothness on the diﬀeomorphisms but try whenever possible to avoid placing such smoothness assumptions on the images themselves. Such a choice, which is very important given the discontinuous nature of images, is made at the cost of increasing technicalities and notation, as will be seen in section 3, in which the basic geometry of the model is presented. Here, we deﬁne the tangent space at a given square integrable image i as an equivalent class for all possible variations resulting from an inﬁnitesimal combination of a deformation (geometry) and of the addition of a square integrable function (photometry), yielding what can be called a morphometrical variation. We then equip it with an inner product and deﬁne from it lengths and energies of curves. This metric is based on the best tradeoﬀ between geometrical and photometrical variations. Still, in this general setting, we show the existence of minimizing geodesics (curves of minimal energy) between any two images. The rest of the paper is devoted to the study of geodesics and their generation

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

19

from initial conditions. The motivation in this study is the possibilities it oﬀers for prototype-based image representation and the generation of image variations and deformations from initial conditions belonging to a vector space. In this context, the geodesic equations are derived under the assumption that the deformed prototype is smooth (H 1 ), but with no restriction on the other endpoint. This is done in section 5.2.2. The obtained evolution equations are then generalized to a form which does not require the smoothness of the initial position and that we conjecture to represent a comprehensive class of image evolutions. The equations, under this form, are studied in section 7, where we prove that they have a unique solution over arbitrary ﬁnite time intervals. Our last result shows the local nonambiguity of this representation, at least in the smooth case: from a smooth prototype, the solutions of the geodesic equations in small time cannot coincide if they have been generated from distinct smooth initial conditions. This is done in section 9. The last section, 10, presents numerical experiments, which illustrate the feasibility of retrieving a target from the initial conditions associated to the minimizing geodesic starting from the template. 2. Notation. For further reference, we present in a single deﬁnition some of the main functional spaces we use throughout the paper. Definition 1. Let k, p ∈ N∗ , l ∈ N, and Ω be a bounded domain of Rk with C 1 boundary. (1) We denote Cc∞ (Ω, Rp ) the space of smooth compactly supported Rp -valued functions on Ω. (2) We denote C l (Ω, Rp ) the set of the restrictions to Ω of the l times continuously diﬀerentiable Rp -valued functions on Rk . Let f ∈ C l (Ω, Rp ). We deﬁne the norm |f |l,∞ by |f |l,∞

∂ |α| f , d · · · ∂xα d

sup |

α1 x∈Ω ∂x1 α, 0≤|α|≤l

where for any α (α1 , . . . , αd ) ∈ Nd∗ we denote |α| αi . (3) We denote C0l (Ω, Rp ) the completion of Cc∞ (Ω, Rp ) for the norm | |l,∞ . (4) We denote L2 (Ω, Rp ) the Hilbert space of square integrable functions in Rp with dot product deﬁned for f, g ∈ L2 (Ω, Rp ) by f, g2 f (x), g(x)Rp dx. Ω

(5) We denote H 1 (Ω, Rp ) the Hilbert space of square integrable Rp -valued functions with square integrable ﬁrst partial (generalized) derivatives. The dot product is deﬁned for any f, g ∈ H 1 (Ω, Rp ) by f, g

H1

f, g2 +

k ∂f i=1

∂g , ∂xi ∂xi

. 2

3. Measuring distances on the image space. 3.1. Inﬁnitesimal transformations. Let us consider a space JW of functions ¯ and taking values on Rd , which will be explicitly deﬁned later. To deﬁned on Ω, somewhat ﬁx the ideas, we shall speak of elements of JW as “images” and use the

20

´ AND LAURENT YOUNES ALAIN TROUVE

corresponding photometric vocabulary, although our constructions apply to generic graphs of vector-valued functions. We want to build a distance, denoted hereafter dJW , on JW through a Riemannian analysis. Let j ∈ JW and h ∈ R, and consider a small perturbation jh of j such that jh (x) = j(x − hv(x)) + hσ 2 z(x) + o(h), where v is a displacement ﬁeld and z is an Rd -valued function on Ω. Here and in the following, σ 2 is a ﬁxed positive parameter. The transformation from j to jh is therefore divided in two complementary processes. The ﬁrst, which we call the “geometric transformation,” is a pure deformation of the support for which a point located at x in the ﬁrst image is pushed to location x + hv(x). The second process, called the “photometric transformation,” is the residual, obtained by the addition of σ 2 hz. Both transformations are the main ingredients of any morphing process between two images. When j is smooth, we have (1)

∂j jh − j = σ 2 z − dj(v). lim ∂h |h=0 h→0 h

∂j is an element of the tangent The usual geometric interpretation is that γ ∂h |h=0 space Tj JW , and, given our representation, it is sensible to let the length |γ|j depend on w (z, v) and to let w vary in some chosen vector space W . The solution cannot merely be to set |γ|j = |w|W , where | |W is a norm on W , because the representation (z, v) → γ is not one-to-one: if w = (v , z ) is such that

σ 2 (z − z) − dj(v − v) = 0,

(2)

then the transformations along w and w of j are inﬁnitesimally equivalent. Hence, looking for the best tradeoﬀ between geometric and photometric transformations, we can choose for the metric on the tangent space Tj JW (3) |γ|j = inf |w|W | w = (v, z), γ = σ 2 z − dj(v) . Now, we can deﬁne formally (4)

dJW (j0 , j1 ) inf 0

1

∂j

, j path from j0 to j1 .

∂t

jt

3.2. Diﬀerentiable structure. The previous construction is now made rigorous for JW L2 (Ω, Rd ). Remark 1. Since L2 (Ω, Rd ) is a Hilbert space, it has a natural structure of smooth inﬁnite dimensional manifold. However, the diﬀerential structure we need to consider here is diﬀerent from the standard L2 structure. To see this, consider the following example: Ω =]0, 1[k , and jh (x) j0 (x − hv(x)), where • j0 (x) 1x1 ≥1/2 , • v ∈ Cc∞ (Ω, Rk ) is such that the ﬁrst coordinate, v1 , of v is strictly positive at the center c (1/2, . . . , 1/2) of Ω. Then, |jh − j0 |2 /h → +∞ so that jh is not diﬀerentiable at h = 0 for the usual L2 diﬀerentiable structure, whereas, by the construction above, it will be so for the differential structure on JW (this is a justiﬁcation for keeping the nonstandard notation JW for the image space). Our construction starts with the deﬁnition of C 1 paths on JW . We ﬁrst need to specify the allowed geometric as well as grey-level inﬁnitesimal transformations.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

21

3.2.1. Inﬁnitesimal transformation spaces. Geometric transformation. We denote B the space of the displacement ﬁelds underlying the inﬁnitesimal geometric transformation. We assume that B is a Hilbert space with dot product denoted by , B and norm denoted by | |B . We assume throughout this paper that B is continuously embedded in C0p (Ω, Rk ), where p = 1 at least but may be larger if speciﬁed. As a reminder, we recall that B is continuously embedded in some Banach space B (with norm | |B ) of functions if and only if each element v of B can be considered as an element of B and there exists a constant C such that, for all v ∈ B, |v|B ≤ C |v|B . Moreover, B is compactly embedded in B if it is continuously embedded and any bounded set for the norm on B is relatively compact in the B -topology. We shall also assume that Cc∞ (Ω, Rk ) is dense in B. Photometric transformation. Grey-level transformations are assumed to belong to the space L2 (Ω, Rd ). Finally, we denote W B × L2 (Ω, Rd ) on which we place the dot product deﬁned for w = (v, z) and w = (v , z ) by w, w W v, v B + σ 2 z, z 2 . 3.2.2. Diﬀerentiable curves and tangent space. For any smooth image j, we have, for any u ∈ Cc∞ (Ω, Rd ) and any w = (v, z) ∈ W , σ 2 z − dj(v), u2 = σ 2 z, u2 + j, div(u ⊗ v)2 ,

(5)

where div(u⊗v) ∈ C0 (Ω, Rd ) is deﬁned by div(u⊗v)i = div(ui v). The right-hand side of the equality is well deﬁned for arbitrary j ∈ JW , which leads us to the following deﬁnition. Definition 2 (C 1 curves in JW ). Let I be an interval in R. We say that j : I → JW is a continuously diﬀerentiable curve if there exists w (v, z) ∈ C(I, W ) such that (1) j ∈ C(I, L2 (Ω, Rd )) for the usual L2 -topology, (2) for any u ∈ Cc∞ (Ω, Rd ), t → jt , u2 is a continuously diﬀerentiable real-valued ∂ function and ∂t jt , u2 = σ 2 zt , u2 + jt , div(u ⊗ vt )2 . If we deﬁne as usual tangent vectors via classes of ﬁrst order equivalent curves, we can identify the tangent bundle of JW from the deﬁnition of C 1 path on JW as follows. Definition 3. (1) For any j ∈ JW and any u ∈ Cc∞ (Ω, Rd ), we denote lj,u the continuous linear form on W (the continuity stems from the continuous embedding of B in C01 (Ω, Rk )) deﬁned for any w = (v, z) ∈ W by (6)

lj,u (w) σ 2 z, u2 + j, div(u ⊗ v)2 .

(2) We deﬁne (7)

Ej { w ∈ W | lj,u (w) = 0, ∀u ∈ Cc∞ (Ω, Rd ) },

and (8)

Tj JW {j} × W/Ej ,

where W/Ej is the quotient space, the elements of which are denoted w.

22

´ AND LAURENT YOUNES ALAIN TROUVE

Remark 2. The use of a quotient space is a consequence of the nonuniqueness of the representation of the derivative by an element w ∈ W as explained by (2). We consider Tj JW as a vector space where for any γ = (j, w) and γ = (j , w ) ∈ Tj JW , we have γ + λ γ (j, w + λw ). Now, if we deﬁne T JW

T j JW ,

j∈JW

T JW plays the role of the tangent bundle of the manifold JW . Definition 4. (1) We denote π : T JW → JW the canonical projection deﬁned by π(γ) = j for any γ (j, w) ∈ Tj JW . (2) Let γ (j, w) ∈ T JW and w = (z, v) ∈ w. For any u ∈ Cc∞ (Ω, Rd ), we denote γ , u σ 2 z, u2 + j, div(u ⊗ v)2 . (Note that the right-hand side does not depend on the choice of w ∈ w). (3) For any function γ : I → T JW where I is a real interval, we say that γ is measurable if π ◦ γ is measurable from I to JW and for any u ∈ Cc∞ (Ω, Rd ), γt , u is measurable from I to R. Returning to Deﬁnition 2, we see that C 1 curves j admit a lifting t → γt = (jt , wt ) to T JW such that for all u ∈ Cc∞ (Ω, Rd ) d jt , u2 = γt , u dt so that it is natural to deﬁne (9)

djt dt

γt ∈ Tjt JW leading to the formula

d jt , u2 = dt

djt ,u . dt

The next step, for our Riemannian construction, is to place a metric on Tj JW for all j ∈ JW . 3.3. Riemannian structure. Definition 5. For any j ∈ JW , we deﬁne on Tj JW the norm |γ|j inf{ |w|W | (j, w) ∈ γ }. The inﬁmum is attained at a unique point, as stated in the following proposition. Proposition 1. For any j ∈ JW and any γ = (j, w) ∈ Tj JW , since w is a closed subspace of W , there exists a unique w ∈ W denoted p(γ) such that p(γ) Argmin |w|W . w∈w

Hence, |γ|j |p(γ)|W . Moreover, p is linear from Tj JW to W . Proof. Since w is a close subspace of W , it is suﬃcient to note that if p is the orthogonal projection from W to Ej⊥ , then p(w) = 0 for any w ∈ Ej so that p can be factorized as a linear map p from W/Ej to Ej⊥ . Now, one easily checks that p(γ) ∈ w and that p(γ) minimizes the norm.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

23

We can now deﬁne the geodesic distance between arbitrary points j0 , j1 in JW by

1

dj 1

dt | j ∈ Cpw dJW (j0 , j1 ) inf (10) ([0, 1], JW ), j0 = j0 , j1 = j1 ,

dt

0

jt

1 where Cpw ([0, 1], JW ) is the set of piecewise C 1 curves in JW which are straightforwardly deﬁned from the deﬁnition of C 1 curves. This deﬁnition is the usual deﬁnition for ﬁnite dimensional Riemannian manifolds. There is, however, a measurability issue, since it is not obvious from our deﬁnition of a measurable path in T JW that t → |γt |π(γt ) is measurable. This issue is addressed in Proposition 2, the proof of which is provided in Appendix A. Proposition 2. Let γ : [0, 1] → T JW be a measurable path in T JW . Then, p ◦ γ is a measurable path in W and |γ|π◦γ is a measurable real-valued function.

4. Groups of diﬀeomorphisms. Curves in W naturally generate diﬀeomorphisms on Ω by integration of their ﬁrst component, which is a time-dependent vector ﬁeld on Ω which vanishes at ∂Ω. The relations between the Hilbert structure on B and the class of diﬀeomorphisms which can be generated in that way have been investigated, in particular, in [30] and [16], in which suﬃcient smoothness conditions on the vector ﬁeld are derived to ensure existence, uniqueness, and smoothness of the ﬂow for all time. For T > 0, deﬁne the set L1 ([0, T ], B) as the Banach space of measurable functions v : [0, T ] → B such that |v|1,T

T

|v|B dt < ∞.

0

Similarly, L2 ([0, T ], B) denotes the Hilbert space of square integrable functions deﬁned on [0, T ] and taking values in B, with the norm

|v|2,T

0

T

1/2 2 |v|B

dt

.

For v ∈ L1 ([0, T ], B), consider the ODE (11)

dy = vt (y). dt

A global ﬂow solution of this equation is a time-dependent family of functions t → ϕt such that, for all x ∈ Ω, ϕ0 (x) = x and

t

vs ◦ ϕs ds.

ϕt = 0

When the dependence of this ﬂow on v must be emphasized, it is denoted by ϕv . Results in [30, 16] essentially relate the existence and smoothness of such ﬂows to embedding conditions of B into standard sets of continuous functions. We quote these results in the following theorem. Theorem 1 (Trouv´e). If B is continuously embedded in C01 (Ω, Rk ), then for all T > 0 and all v ∈ L1 ([0, T ], B), the ODE (11) can be integrated over [0, T ], and its associated ﬂow ϕv is such that at all times x → ϕvt is a homeomorphism of Ω.

24

´ AND LAURENT YOUNES ALAIN TROUVE

Notation 1. Assume that B is continuously embedded in C01 (Ω, Rk ), and introduce the map AT : L1 ([0, T ], B) → C(Ω, Rk ), v → ϕvT . Then, the set A1 (L1 ([0, 1], B)) will be denoted GB . The fact that GB is a group is proved in [30]. Further results on these groups and on AT can be found in Appendix C. The relation between algebraic and metric properties of groups of diﬀeomorphisms and some of the fundamental equations of ﬂuid mechanics has been the subject of several studies, starting with [5], in which the Euler equation is related to the geodesic equations of groups of diﬀeomorphisms with an L2 metric on its Lie algebra (see also [3, 4, 24]). Another important equation, the Camassa–Holm equation, which describes the motion of the waves in shallow water, can be interpreted along the same lines with an Hα1 metric on the Lie algebra [11, 17]. Here, since the energy derives from both geometric and photometric variations, the geodesic equations that we derive can be formally interpreted as conservation of momentum on a semidirect product of the group of diﬀeomorphisms and the space of images, as studied in [25]. However, our point of view of smooth deformations acting on nonsmooth images requires a speciﬁc approach. This is also related to developments in optimal design [28]. 5. Geodesics on JW . 5.1. Minimizing geodesics. The space of C 1 curves is not well suited to deal with proofs of the existence of curves of minimal length between two images j0 and j1 , i.e., minimizing geodesics. We introduce below the more tractable space of curves with square integrable speed. We need ﬁrst a preliminary proposition saying that square integrable paths in T JW are uniquely identiﬁed by their trace on smooth space-time vector ﬁelds in Rd . The proof of this proposition is postponed to Appendix A. Proposition 3. Let γ : [0, 1] → T JW be a measurable path in T JW . Then, if 1 1 2 ∞ d |γ t |π(γt ) dt < +∞ and, for any u ∈ Cc (Ω×]0, 1[, R ), we have 0 γt , ut dt = 0, 0 then γ = 0 a.e. We can now introduce the space H 1 ([0, 1], JW ) of regular curves. Definition 6. We say that a path j ∈ C([0, 1], L2 (Ω, Rd )) is regular if there exists 1 a measurable path γ : [0, 1] → T JW such that π(γ) = j, 0 |γt |2 dt < ∞, and, for any 1 1 u ∈ Cc∞ (]0, 1[×Ω, Rd ), we have − 0 jt , ∂u ∂t 2 dt = 0 γt , ut dt. From Proposition 3, ∂j γt , we get the integration by the path γ is uniquely deﬁned; using the notation ∂t parts formula 1 1 ∂j ∂u (12) jt , dt = − , ut dt. ∂t 2 ∂t 0 0 We denote H 1 ([0, 1], JW ) as the set of all the regular paths in C([0, 1], L2 (Ω, Rd )). Proposition 4. We have C 1 ([0, 1], JW ) ⊂ H 1 ([0, 1], JW ) and both deﬁnitions of ∂j ∂t coincide. Proof. Let j ∈ C 1 ([0, 1], JW ). There exists w = (v, z) ∈ C([0, 1], W ) such that for any u ∈ Cc∞ (Ω, Rd ), t → jt , u2 is C 1 and ∂ jt , u2 = σ 2 zt , u2 + jt , div(u ⊗ vt )2 . ∂t

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

25

Certainly, w ∈ L2 ([0, 1], W ). Moreover, for any u ∈ Cc∞ (Ω, Rd ) and any f ∈ Cc∞ (]0, 1[, R), we have by integration by parts (we denote f (t) df dt ) 1 1 1 d jt , f (t)u2 = f (t)jt , u2 dt = − f (t) jt , u2 dt dt 0 0 0 so that (12) is true for u ⊗ f ∈ Cc∞ (]0, 1[×Ω, Rd ). The complete proof follows by usual density arguments. We carry on with an important result which characterizes regular paths in JW . For a path v in L1 ([0, 1], B), we deﬁne for any s, t ∈ [0, 1] ϕvt,s ϕvs ◦ (ϕvt )−1 . Theorem 2. A path j : [0, 1] → JW is regular (resp., is in C 1 ([0, 1], JW )) if and only if there exists w = (v, z) ∈ L2 ([0, 1], W ) (resp., ∈ C([0, 1], W )) such that t jt = j0 ◦ ϕvt,0 + σ 2 zs ◦ ϕvt,s ds. 0

Proof. The proof is postponed to Appendix B. Theorem 3. Let j0 and j1 be in JW . Then we have

1

∂j

1

(13) dJW (j0 , j1 ) = inf

∂t dt | j ∈ H ([0, 1], JW ), j0 = j0 , j1 = j1 . 0 jt Proof. Let j ∈ H 1 ([0, 1], JW ) be a regular path from j0 to j1 and let w ∈ ∂j L ([0, 1], W ) such that wt = pjt ( ∂t ) for any t. There exists a sequence (wn = 1 (vn , zn ) ∈ C([0, 1], W ), n ∈ N) such that 0 |wt − wtn |2W dt → 0. Deﬁne t n n zns ◦ ϕvt,s ds. jnt = j0 ◦ ϕvt,0 + σ 2 2

0

We get from Theorem 2 that jn ∈ C 1 ([0, 1], JW ). Now, considering w ˜ n (˜ vn , ˜zn ) n n n vn n n with ˜zt zt + (j1 − j1 ) ◦ ϕs,1 and v ˜ v we get from Theorem 9 (see Appendix C) that w ˜ n ∈ C([0, 1], W ). Using Theorem 2, we deduce that if ˜jn is deﬁned by t n n ˜zns ◦ ϕvt,s ds, ˜jnt = j0 ◦ ϕvt,0 + σ 2 0

then ˜j ∈ C ([0, 1], JW ) and = j1 . However, 1 n

1 1

∂˜j

n

dt ≤

| w ˜ | dt → |wt |W dt t W

∂t n 0 0 0 ˜j n

1

˜jn1

t

1 ∂j when n → ∞. Therefore, we deduce that dJW (j0 , j1 ) ≤ 0 | ∂t |jt dt for any regular path from j0 to j1 . Finally, since C 1 ([0, 1], JW ) ⊂ H 1 ([0, 1], JW ), we get the result. Definition 7. Let j0 , j1 ∈ JW . We say that j ∈ C([0, 1], L2 (Ω, Rd )) is a minimizing geodesic path from j0 to j1 if j is regular and

12 1

2

∂j dt = dJW (j0 , j1 ).

∂t

0 jt We denote GJW (j0 , j1 ) as the set of the minimizing geodesic paths from j0 to j1 .

´ AND LAURENT YOUNES ALAIN TROUVE

26

5.2. Characterization of geodesics. 5.2.1. Photometric optimality. Theorem 4. Let j0 , j1 ∈ JW and j ∈ GJW (j0 , j1 ) be a minimizing geodesic path from j0 to j1 . Let w = (v, z) ∈ L2 ([0, 1], W ) be deﬁned by wt p( ∂j ∂t ) for any t ∈ [0, 1]. Then z ∈ C([0, 1], L2 (Ω, Rd )) and for any t ∈ [0, 1] we have

zt = z0 ◦ ϕvt,0 dϕvt,0 .

(14)

Proof. Let j ∈ H 1 ([0, 1], JW ) be a minimizing geodesic from j0 to j1 , and let dj w = (v, z) ∈ L2 ([0, 1], W ) such that for any t ∈ [0, 1], wt = p( dt ). For any u ∈ Cc∞ (]0, 1[×Ω, Rd ) and any ε ∈ R, deﬁne t ˜jt = j0 ◦ ϕvt,0 + σ 2

zs + ε 0

∂us ◦ ϕvs,1 ∂s

◦ ϕt,s ds.

v 2 t Since t → (vt , zt + ε ∂u j ∈ ∂t ◦ ϕt,1 ) ∈ L ([0, 1], W ), we get from Theorem 2 that ˜ 1 H ([0, 1], JW ). Moreover, ˜j0 = j0 and ˜j1 = j1 so that

0

1

2

djt

dt =

dt

1

0

jt

2 |vt |B

+σ

2

2 |zt |2

2

d˜jt

dt

dt

˜jt

1

dt ≤ 0

1

≤

2 |vt |B

0

2

∂ut v

◦ ϕt,1 dt. + σ z t + ε ∂t 2

2

Since ε is arbitrary, we get 0= 0

1

∂ut ◦ ϕvt,1 zt , ∂t

zt ◦

dt = 2

1

0

ϕv1,t

v ∂ut

dϕ1,t , ∂t

dt. 2

d 2 d Choosing arbitrary u ∈ Cc∞ (]0, 1[×Ω,

R ), we get that there exists z˜1 ∈ L (Ω, R ) such that t-a.e. we have zt ◦ ϕv1,t dϕv1,t = z˜1 . Hence, if ˜zt = z˜1 ◦ ϕvt,1

dϕvt,1 , we have ˜z ∈ C([0, 1), L2 ([0, 1], Rd )) and zt = ˜zt t-a.e. Note that ˜z0 ◦ ϕv1,0 dϕv1,0 = z˜1 so that

˜zt = ˜z0 ◦ ϕv1,0 dϕv1,0 ◦ ϕvt,1 dϕvt,1 = ˜z0 ◦ ϕvt,0 dϕvt,0 , and the proof is ended. This leads to the following deﬁnition. Definition 8. A regular path j ∈ H 1 ([0, 1], JW ) is called a pregeodesic path if and only if the following equations are satisﬁed almost everywhere in t:

(15)

⎧ t ⎪ v 2 ⎪ jt = j0 ◦ ϕt,0 + σ ⎪ zs ◦ ϕvt,s ds, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨

zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dj ⎪ ⎪ . ⎩ (vt , zt ) = p dt

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

27

5.2.2. Study of the geodesic equation. Directional derivatives in L2 . In this section, we try to clarify the last equation of system (15), at least in some situations of interest. The diﬃculty comes from the fact that, unless jt is smooth enough, this equation does not, in general, specify a unique correspondence zt → vt . To be more precise, let us analyze the condition that, for all t, dj . (vt , zt ) = p dt For this purpose, we ﬁrst introduce a weak version of the directional derivative dj.v when j ∈ JW and v ∈ B. Definition 9. Let j ∈ JW . (1) We deﬁne the operator Dj : Dj → L2 (Ω, Rd ) by Dj { v ∈ B | ∃C, s.t. ∀u ∈ Cc∞ (Ω, Rd ), |j , div(u ⊗ v)2 | ≤ C |u|2 }, and for any v ∈ Dj , Dj.v is the unique element in L2 (Ω, Rd ) such that Dj.v , u2 = −j , div(u ⊗ v)2 Cc∞ (Ω, Rd ).

for any u ∈ (2) We deﬁne the adjoint operator Dj ∗ : D∗j → B, where D∗j { u ∈ L2 (Ω, Rd ) | ∃C, s.t. ∀v ∈ Dj |Dj.v , u2 | ≤ C |v|B }, and, for any u ∈ D∗j , Dj ∗ is the unique element in Dj (closure of Dj ) such that Dj ∗ .u , vB = u , Dj.v2

(16)

for any v ∈ Dj . Remark 3. The existence of Dj.v comes from the extension of the linear form u → j , div(u ⊗ v) for smooth u into a continuous linear form on L2 (Ω, Rd ) for v ∈ Dj . For the deﬁnition of the adjoint Dj ∗ , the adjoint is uniquely deﬁned as an element of Dj by (16) (Dj is not necessarily dense in B). Fix j ∈ JW . We may characterize elements v ∈ Dj as follows. (We denote hereafter ϕv. as the ﬂow associated with the constant speed vt ≡ v for any t ∈ [0, 1].) Theorem 5. The vector ﬁeld v ∈ B belongs to Dj if and only if there exists a square integrable function ξ : Ω → Rd such that t

−1

−1

(17) ξ ◦ ϕv0,s (x) dϕv0,s (x) ϕvs,t ds. j ◦ ϕv0,t (x) = j(x) dx ϕv0,t + 0

We have in such a case Djv = ξ − jdiv(v). Proof. We ﬁrst notice that, if v ∈ B,

d d v v

j(x) , u ◦ ϕε,0 (x) dx ϕε,0 dx = j ◦ ϕv0,ε (x) , u(x) dx. −j , div(u ⊗ v)2 = dε Ω dε Ω Assuming that (17) holds, the last expression yields ε

−1

−1

d d

j(x) , u(x) dx ϕv0,ε dx + ξ ◦ ϕv0,s , u(x) dϕv0,s (x) ϕvs,ε dx dε Ω dε Ω 0 =− j(x) , u(x)div(v)dx + ξ(x) , u(x)dx = ξ − jdiv(v) , u2 , Ω

Ω

´ AND LAURENT YOUNES ALAIN TROUVE

28

which implies that v ∈ Dj and Djv = ξ − jdiv(v). Conversely, let v ∈ Dj and ξ = Djv+ jdiv(v). Fix u ∈ C 1 (Ω, Rd ). Consider the function f , deﬁned on [0, 1] by f (t) = j ◦ ϕv0,t , u 2 . Denote by ˜j(t) the left-hand term of (17), and g(t) = ˜jt , u2 . We have

−1

g (t) = − j , udivϕv0,t (x) v dx ϕv0,t

+ ξ ◦ ϕv0,t , u 2 2 t

−1

ξ ◦ ϕv0,s dϕv0,s ϕvs,t , udivϕv0,t v ds − 0 2 v = ξ ◦ ϕ0,t − ˜jt divϕv0,t v , u . 2

Since f (t + ε) = j ◦ ϕvt,t+ε , u ◦ ϕvt,0 dϕvt,0 2 , we have

f (t) = Djv , u ◦ ϕvt,0 dϕvt,0 2 = (Djv) ◦ ϕv0,t , u 2 . Therefore, computing the integral of the diﬀerence and using the deﬁnition of ξ,

j ◦ ϕv0,t − ˜jt , u

t

2

= 0

j ◦ ϕv0,s − ˜js , udivϕv0,t v

2

ds ≤ |u|2 |v|B

0

t

j ◦ ϕv0,s − ˜js ds. 2

Taking the supremum of the left-hand term over continuously diﬀerentiable u with L2 -norm equal to 1 yields

j ◦ ϕv0,t − ˜jt ≤ |v| B 2

t

0

j ◦ ϕv0,s − ˜js ds, 2

which implies j ◦ ϕv0,t − ˜jt 2 = 0 for all t. An interesting consequence of this is the following lemma. Lemma 1. For any j ∈ L2 (Ω, Rd ), one has j ∈ Dj∗ and, for v ∈ Dj , Djv , j2 = −

1 2 |j| , divv . 2 2

Proof. Indeed, let v ∈ Dj . Consider the function

j ◦ ϕv0,t (x) 2 dx. f (t) = Ω

2 Since f (t) = |j|2 , |dϕt,0 | 2 , we have f (0) = −|j| , divv2 . Using, on the other hand, expression (17) yields f (0) = 2Djv , j2 . Interpretation of the pregeodesic equations. The property that w = (v, z) ∈ W belongs to Ej , which states that, for all u ∈ Cc∞ (Ω, Rd ), σ 2 z, u2 + j, div(u ⊗ v)2 = 0 is equivalent to v ∈ Dj and σ 2 z − Dj.v = 0. Consider now some tangent vector γ ∈ Tj JW , and study the property that, for some w = (v, z) ∈ W , one has p(γ) = w. This implies, in particular, that, for all (v , z ) ∈ Ej , |v + v |B + σ 2 |z + z |2 ≥ |v|B + σ 2 |z|2 , 2

2

2

2

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

29

which is in turn equivalent to the following: for all v ∈ Dj , v , v B +z , Dj.v 2 = 0. This implies that z ∈ D∗j and that v , v B +Dj ∗ z , v B = 0, so that v = −Dj ∗ z+γ⊥ , where γ⊥ is the projection of v onto D⊥ j . Note that this orthogonal component does not depend on the choice of (v, z) from the equivalence class deﬁning γ (hence the notation). We thus may conclude that (v, z) = p(γ) if and only if z ∈ D∗j and v = −Dj ∗ z + γ⊥ . The ﬁrst conclusion we may draw from this is that, whenever Dj is dense in B, dj vt is uniquely determined by zt and the condition (vt , zt ) = p( dt ). It is given by vt = −Dj∗t zt . This is true, for example, when jt ∈ H 1 (Ω, Rd ) at all times, since, in this case Djt = B (notice that, by Theorem 4, this is true along a geodesic as soon as j0 and j1 belong to H 1 (Ω, Rd )). However, this is not the general situation. As an example, consider the case when j is the indicator function of a subdomain Ω1 of Ω with smooth boundary. If v is a vector ﬁeld on Ω and u is a smooth function on Ω, we have j , divuv = uv , ν1 Rk dσ1 , ∂Ω1

where ν1 is the outward normal to ∂Ω1 and σ1 is the surface measure on ∂Ω1 . This implies that djv may be identiﬁed to a singular measure supported to ∂Ω1 , which does not belong to L2 unless it vanishes. Thus, Dj consists exactly of vector ﬁelds on Ω which belong to B and have vanishing normal components on ∂Ω1 . For a ∈ Rk and x ∈ R, denote by Kx a the element of B such that Kx a , uB = u(x) , aRk . Then, Kx ν(x) belongs to Dj⊥ for any x ∈ ∂Ω, and so does any linear combination of these vector ﬁelds. We see that in this case Dj⊥ is nontrivial. This discussion implies that the pregeodesic condition for a path may be written ⎧ t ⎪ v 2 ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, ⎪ t 0 t,0 ⎪ ⎪ 0 ⎪ ⎨

⎪ zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt ∈ Dj∗ , and vt − Dj∗t zt ∈ Dj⊥t .

(18)

These equations are not complete yet (in the sense that they cannot be solved from the initial values (j0 , v0 , z0 )) since they provide no information on the choice of vt − Dj∗t zt at time t (unless of course Dj⊥t = {0}). We need to specify the mode of propagation of this singular component along a geodesic. The following computation provides a hint on possible ways to achieve this. Assume that j is pregeodesic and dj (vt , zt ) = p( dt ). In such a case, we have 2

|zs |2 =

|z0 |2 |dϕv0,s |−1 dy

|z0 ◦ ϕvs,0 |2 |dϕvs,0 |2 dx = Ω

Ω

and z0 (x) =

1 (j1 ◦ ϕv0,1 (x) − j0 (x)) σ2

0

1

|dϕv0,s |−1 ds.

´ AND LAURENT YOUNES ALAIN TROUVE

30 Thus

1

0

2 |zs |2

1 ds = 4 σ

2

j1 ◦ ϕv0,1 − j0

. 1 |dϕv0,s |−1 ds Ω 0

Making the change of variables y = ϕv0,1 (x) yields 0

1

2 |zs |2

j1 − j0 ◦ ϕv1,0 2

1 ds = 4 σ

Ω

1 0

|dϕv1,0 (x) ϕv0,s |−1 |dϕv1,0 (x)|−1 ds

,

i.e.,

1

0

2 |zs |2

1 ds = 4 σ

2

j1 − j0 ◦ ϕv1,0

, 1 |dϕv1s |−1 ds Ω 0

and the geodesic energy is given by (19) 0

1

2 |vs |B

1 ds + 2 σ

2

j1 − j0 ◦ ϕv1,0

. 1 |dϕv1,s |−1 ds Ω 0

We can obtain more precise information on the geodesic by studying variations of this expression with respect to v. This will be handled below, under a smoothness assumption on j0 . Before this, we need some notation for the reproducing kernel on B. They will be useful throughout the paper. Kernels for the inner-product on B. Proposition 5. There exists a continuous operator K (resp., K∇ ) on L1 (Ω, Rk ) (resp., L1 (Ω, R)) with values in B such that, for all u ∈ L1 (Ω, Rk ) (resp., u ∈ L1 (Ω, R)), for all v ∈ B, Ku , vB = u, v2 , and K∇ u , vB = −u, divv2 . Proof of Proposition 5. Let u ∈ L1 (Ω, Rk ). Since we assume that B is continuously embedded in C01 , the linear form deﬁned on B by v → u, v2 is continuous because |u, v2 | ≤ |u|1 |v|∞ . Therefore, there exists a unique element in B, denoted Ku, such that, for all v ∈ B, Ku , vB = u, v2 and continuity comes from the inequality Ku , vB ≤ |u|1 |v|∞ ≤ cst |u|1 |v|B . The same proof holds for K∇ , since |divv|∞ is also controlled by |v|B . It can be remarked that, for smooth u, K∇ u = K(∇u). Remark 4. When j is smooth (e.g., j ∈ H 1 (Ω, Rd )), the operator Dj ∗ introduced in the previous paragraph is given by Dj ∗ z = K(dj ∗ .z), in which dj ∗ is the standard matrix adjoint of dj. Indeed, we have in this case z , Dj.v2 = z , dj.v2 = dj ∗ .z , v2 = K(dj ∗ .z) , vB .

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

31

Characterization with a smooth endpoint. We study the eﬀect of small variations in v on the geodesic energy (19), under the additional hypothesis that j0 ∈ H 1 (Ω, Rd ). Thus, ﬁx h ∈ L2 ([0, 1], B), and consider a perturbation v + εh of v. We compute the corresponding variation of the geodesic energy. The variation of the 1 ﬁrst term being 2 0 vt , ht B dt, we can focus on the second term, namely, 1 U 2 σ ε

2

j0 ◦ ϕv+εh − j1

1,0 dx. 1 v+εh −1 |dϕ | ds Ω 1s 0

The variations of U ε are given in Lemma 2, which is proved in Appendix E. Lemma 2. We have, at ε = 0, (20) 1 dU ε 2 2 2 =σ K∇ (qtv |zt | ) + K(|zt | ∇qtv ) + 2K([dϕvt,0 ]∗ dϕvt,0 j0∗ zt ), ht dt, dε B 0 t with qtv = 0 |dϕvt,s |−1 ds. We can deduce from this our additional condition for a regular path to be a minimizing geodesic: for almost all t ∈ [0, 1], vt +

1 (KDtv + K∇ Ctv )B = 0, 2

where Dtv σ 2 |zt | ∇qtv + 2[dϕvt,0 ]∗ dϕvt,0 j0∗ zt , 2

and 2

Ctv σ 2 qtv |zt | . It may be interesting to check that this condition boils down to the one we have obtained before for smooth trajectories, namely, vt + K(djt∗ zt ) = 0. It suﬃces to notice that, for pregeodesic trajectories, jt = j0 ◦ ϕvt,0 + σ 2 zt qtv and that, when zt is smooth, KDtv + K∇ Ctv = K(Dtv + ∇Ctv ). We now deﬁne geodesic paths (not necessarily minimizing). Definition 10. Let j0 ∈ H 1 (Ω, Rd ). A regular path j ∈ H 1 ([0, 1], JW ) starting at j0 is called a geodesic path if and only if there exists w = (v, z) ∈ L2 ([0, 1], W ) such that the following equations are satisﬁed almost everywhere in t: t ⎧ v 2 ⎪ ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, t 0 ⎪ t,0 ⎪ ⎪ 0 ⎪ ⎪ ⎨

(21) zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎩ v + K([dϕv ]∗ d v j ∗ z ) + σ K (q v |z |2 ) + K(|z |2 ∇q v ) = 0, t ϕt,0 0 t ∇ t t t t,0 t 2

32

´ AND LAURENT YOUNES ALAIN TROUVE

t with qtv = 0 |dϕvt,s |−1 ds. These equations are complete; it will be shown in section 7 that initial conditions (j0 , z0 ) uniquely specify the solutions. It is interesting to check that geodesics as deﬁned in (21) also are pregeodesics. For this, we ﬁrst show that, for all times t, zt ∈ D∗jt . Noting that the ﬁrst equation in (21) may also be written jt = j0 ◦ ϕvt,0 + σ 2 zt qt it is clear that Djt = Dzt , and zt ∈ Dz∗t is proved in Lemma 1. The same lemma also provides the fact that, for w ∈ Dzt , σ2 2 2 |zt | , div(qtv w) , zt , Djt w2 = zt , d(j0 ◦ ϕvt,0 )w 2 + σ 2 |zt | ∇qt , w − 2 2 and this is equal to −vt , wB by deﬁnition of K and K∇ . We thus obtain the fact that vt + Djt∗ zt ∈ Dj⊥t as required. We shall prove existence of solutions for a broader class of evolution equations, extending the range of initial values v0 . Consider the term ut = K([dϕvt,0 ]∗ dϕvt,0 j0∗ z0 ◦

ϕvt,0 dϕvt,0 ) which appears in the third equation of (21). We have, letting ω0 = −dj0∗ z0 , and, for w ∈ B,

ut , wB = [dϕvt,0 ]∗ dϕvt,0 j0 z0 ◦ ϕvt,0 dϕvt,0 , w 2 L

v

v v

= dϕvt,0 j0 z0 ◦ ϕt,0 dϕt,0 , dϕt,0 w L2 v −1 v = ω0 , (dϕ0,t ) w ◦ ϕ0,t L2 . We know, by Appendix C, that ϕv0,t belongs to C p (Ω) as soon as B is continuously embedded in C0p (Ω, Rk ), which implies in this case (with p ≥ 1) that

(dϕv0,t )−1 w ◦ ϕv0,t

p−1,∞

≤ Const |w|B ,

the constant depending on |v|1,B . But this implies in turn that, if the L2 inner product is replaced by the action of any continuous functional, ω0 , on C0p−1 (Ω, Rk ), which will be denoted

ω0 , (dϕv0,t )−1 w ◦ ϕv0,t ,

there exists an element of B that we shall still denote ut such that ut , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t . With this notation, we may formulate the following deﬁnition. Definition 11. Let j0 ∈ L2 (Ω, Rd ). Let ω0 be a continuous linear functional on p−1 (Ω, Rk ) and z0 ∈ L2 (Ω, Rd ). A regular path j ∈ H 1 ([0, 1], JW ) starting at j0 with C initial direction (ω0 , z0 ) is called a generalized geodesic if and only if, for all u ∈ Dj0 , one has (ω0 , u) + z0 , Dj0 u = 0,

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

33

and there exists w = (v, z) ∈ L2 ([0, 1], W ) such that the following equations are satisﬁed almost everywhere in t: ⎧ t ⎪ v 2 ⎪ ⎪ j = j ◦ ϕ + σ zs ◦ ϕvt,s ds, t 0 ⎪ t,0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ zt = z0 ◦ ϕv dϕv , ⎨ t,0 t,0 (22) ⎪ ⎪ ⎪ σ2 2 2 ⎪ v v v ⎪ v K − u + (q |z | ) + K(|z | ∇q ) = 0, t ∇ t t ⎪ t t t ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∀w ∈ B uv , w = ω , (dϕv )−1 w ◦ ϕv 0 t 0,t 0,t B t with qtv = 0 |dϕvt,s |−1 ds. Recall that when j0 is smooth, the only choice is ω0 = dj0∗ z0 , and if z0 is also smooth, the system may be written under the simple form ⎧ t ⎪ v 2 ⎪ jt = j0 ◦ ϕt,0 + σ zs ◦ ϕvt,s ds, ⎪ ⎪ ⎪ 0 ⎪ ⎨

(23) zt = z0 ◦ ϕvt,0 dϕvt,0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vt + σ 2 K(djt∗ zt ) = 0. As an example of the nonsmooth applications we have in mind, assume that j0 is a binary, plane image, which is the indicator function of the interior of a connected open subset Ω1 of Ω with smooth boundary ∂Ω1 . We have seen that any element w ∈ Dj0 should be tangent to ∂Ω1 and that in this case Dj0 w = 0 and D∗j0 = L2 (Ω, R). We therefore may choose z0 arbitrarily in L2 , and (ω0 , w) should vanish for w ∈ Dj0 , which is true, for example, when ω0 is deﬁned by (ω0 , w) = w , ν1 dσ1 , ∂Ω1

where ν1 is the outward normal to ∂Ω1 and σ1 is the surface measure on ∂Ω1 . 6. Existence of minimizing geodesics. The next theorem states that minimizing geodesics always exist between two elements of JW . Theorem 6. Assume that B is compactly embedded in C01 (Ω, Rd ), and let j0 , j1 ∈ JW . Then GJW (j0 , j1 ) is nonempty. Proof. Let (jn )n∈N be a minimizing family of paths in H 1 ([0, 1], JW ) from j0 to j1 ; n for any n ∈ N, let wtn p( djdt ) so that (wn )n∈N is a bounded sequence in L2 ([0, 1], W ). Up to the extraction of a subsequence, we can assume that wn converges weakly to a w∞ in L2 ([0, 1], W ). By lower semicontinuity, we have 1 |wt∞ |2W dt ≤ dJW (j0 , j1 ). 0

By a time change argument, which is classical in the proof that minimizing geodesics travel at constant speed (see [12]), we may furthermore assume that |wtn |W is uniformly bounded by, say, dJW (j0 , j1 ) + 1. Denoting wn = (vn , zn ), consider 2 t ∞ ∞ jt j0 ◦ ϕ∞ z ◦ ϕ∞ is the ﬂow associated to v∞ . Since j t,s ds, where ϕ t,0 + σ 0 s

´ AND LAURENT YOUNES ALAIN TROUVE

34

is a regular path, it is suﬃcient to prove that j1 = j1 . However, if ϕn denotes the ﬂow associated with vn , we know, from Theorem 9, that ϕn1,0 converges uniformly n ∞ 2 d ∞ d to ϕ∞ 1,0 so that j0 ◦ ϕt,0 → j0 ◦ ϕt,0 in L (Ω, R ). Now, let u ∈ Cc (Ω, R ). We

1 n 1 have 0 zs ◦ ϕn1,s , u2 ds = 0 zsn , u ◦ ϕns,1 dϕns,1 2 ds. Since u has bounded derivatives and using Theorem 9 implies the uniform convergence of ϕns1 to ϕ∞ s1 and the pointwise convergence of the derivatives (because of the uniform boundedness of |vsn |B ), we have

1

zns , u

(24) 0

◦

ϕns,1

n

dϕs,1 2 ds −

1

∞

zns , u ◦ ϕ∞ s,1 dϕs,1 2 ds → 0.

0

Moreover, from the weak convergence of zn to z∞ , we get

1

(25) 0

∞

zns , u ◦ ϕ∞ s,1 dϕs,1 2 ds →

1

∞

∞

z∞ s , u ◦ ϕs,1 dϕs,1 2 ds,

0

so that ﬁnally j1 − j1 , u2 = 0 for any u ∈ Cc∞ (Ω, Rd ). Hence j ∈ H 1 ([0, 1], JW ) and the result is proved. 7. Initial value problem for the geodesic equation. We have the following theorem. Theorem 7. Assume that B is continuously embedded in C0p (Ω, Rp ) for p ≥ 3. Then, for all T > 0, there exists a unique solution (v, j, z) of (21) over [0, T ], with initial values j0 ∈ H 1 (Ω, Rd ), z0 ∈ L2 (Ω, Rd ), and ω0 ∈ C p−1 ([0, 1], Rk ) (where C p−1 ([0, 1], Rk ) denotes the topological dual of C p−1 ([0, 1], Rk ) with the norm |ω| sup|v|p−1,∞ ≤1 (ω, v)) which continuously depends on these initial conditions. Continuity of the solution (v, j, z) as a function of (j0 , z0 ) is meant according to H 1 (Ω, Rd ) × L2 (Ω, Rd )-norms for the initial conditions, L2 ([0, T ], W )-norm for (v, z), and C([0, 1], L2 (Ω, Rd ))-norm for j. 8. Proof of Theorem 7. To prove Theorem 7, we show the existence of solutions for short time and then extend them to all time. Fix T > 0. For a given v ∈ L2 ([0, T ], B), let Ψ(v) ∈ L2 ([0, T ], B) be deﬁned by

(26)

⎧ σ2 2 2 ⎪ ⎪ Ψ(v)t = uvt − K∇ (qtv |zvt | ) + K(|zvt | ∇qtv ) , ⎪ ⎪ 2 ⎪ ⎨

v v

v

dϕ , = z ◦ ϕ z 0 ⎪ t t,0 t,0 ⎪ ⎪ ⎪ ⎪ ⎩ v ut , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t .

To estimate the Lipschitz coeﬃcient of Ψ, we introduce v, v ∈ L1 ([0, T ], B) and compute the variation of each term in Ψ(v)t − Ψ(v )t . Fix w ∈ B with |w|B = 1. We have σ2 v 2 v σ2 v 2 |zt | , qt div(w) − |zt | , dqtv w + ω0 , (dϕv0,t )−1 w ◦ ϕv0,t 2 2 2 2 2

σ −1 2 |z0 | , dϕv0,t qtv ◦ ϕv0,t div(w) ◦ ϕv0,t = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t + 2 2 σ 2 2

v

−1 v v |z0 | , dϕ0,t − dϕv0,t qt w ◦ ϕ0,t . 2 2

Ψ(v)t , wB = (27)

35

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

We have qtv

◦

ϕv0,t (x)

t

t

−1

v

v v

v

=

dϕ0,t (x) ϕt,s ds =

dϕ0,s (x) ϕs,t ds = 0

v and letting ξs,t =

0

dx ϕv0,t

dx ϕv ds, 0,s

|dx ϕv0,t | and λvt (w) = (dϕv0,t )−1 w ◦ ϕv0,t , |dx ϕv0,s |

(28) Ψ(v)t , wB =

0

t

σ2 2

t

−1

−1 v

2 |z0 | , ( dϕv0,s div(w) ◦ ϕv0,t − dϕv0,t

∇ξs,t , λvt (w) ) ds

0

+(ω0 , λvt (w)). This implies σ2 2 |Ψ(v )t − Ψ(v)t |B ≤ |z0 |2 sup 2

σ2 2 |z0 |2 sup + 2

t 0

t

v −1

dϕ0,s div(w) ◦ ϕv0,t

v −1 v − dϕ0,s div(w) ◦ ϕ0,t ds : |w|B = 1

v −1 v

dϕ0,t

∇ξs,t , λvt (w)

0

v −1 v v − dϕ0,t

∇ξs,t , λt (w) ds : |w|B = 1

+ |ω0 | sup λvt (w) − λvt (w)

: |w|B = 1 . p−1,∞

The problem is thus reduced

−1 to the estimation of variations, with respect to v, of v λvt (w), ∇ξs,t and of dϕv0,s div(w) ◦ ϕv0,t . They involve diﬀerentials of ϕv , ϕv , and w up to the second degree. The inclusion of B in C 3 ([0, 1], Rk ) and an application of Lemmas 7 and 11 in the appendix directly lead to the estimate C max(|v|1,T ,|v | ) 2 1,T , (29) |Ψ(v)t − Ψ(v )t |B ≤ C σ 2 |z0 |2 + |ω0 | |v − v |1,T e and ﬁnally √ C max(|v|1,T ,|v | ) 2 1,T |Ψ(v) − Ψ(v )|2,T ≤ C T σ 2 |z0 |2 + |ω0 | |v − v |1,T e √ C T max(|v|2,T ,|v | ) 2 2,T . (30) ≤ CT σ 2 |z0 |2 + |ω0 | |v − v |2,T e Therefore, Ψ is q-Lipschitz with q < 1 for T small enough, and its unique ﬁxed point yields a unique solution of (21). This is stated below. Lemma 3. There exists a time T > 0 depending only on |z0 |2 and |j0 |H 1 such that a unique solution of (21) exists on [0, T ]. We now show that this solution can be extended to all times. For this, we prove that there exists a unique ﬁxed point for Ψ at all times. Denote by ΨT this mapping when deﬁned on L2 ([0, T ], B). Clearly, if v is a ﬁxed point of ΨT , its restriction to [0, S] is a ﬁxed point of ΨS . Thus, if T0 is the largest T such that ΨS has a unique ﬁxed point vS in L2 ([0, S], B) for any S < T , then each vT for T < T0 is an extension of vS whenever S ≤ T . We can show that T0 = ∞ by showing that, if T0 < ∞, then

´ AND LAURENT YOUNES ALAIN TROUVE

36

there exists ε > 0 (depending only on T0 and the initial conditions) such that, for all T < T0 , there exists a unique extension of vT to [T, T + ε]. Fix such a T ; the issue of extending a ﬁxed point of ΨT on [T, T + ε] can be rephrased as a ﬁxed point problem for small time with the following notation. For v ∈ L2 ([0, T ], B) and v ∈ L2 ([0, ε], B), deﬁne v ∨ v ∈ L2 ([0, T + ε], B), equal to v on [0, T ] and equal to (t → v (t − T )) on [T, T + ε]. Introduce the function Ψε : L2 ([0, ε], B) → L2 ([0, ε], B) deﬁned by Ψε (v)(t) = ΨT +ε (vT ∨ v)(t − T ). For t > T , qtv

T

∨v

T

v = q1v + qt−T ,

zt = zT ◦ ϕvT t |dϕvT t |−1 , and

uvt , wB = ω0 , (dϕv0,t )−1 w ◦ ϕv0,t = ω0 , (dϕv0,T )−1 (dϕv0,T ϕvT t )−1 w ◦ ϕvT t ◦ ϕv0,T = ωT , dϕvT t )−1 w ◦ ϕvT t with (ωT , w) = ω0 , (dϕv0,T )−1 w ◦ ϕv0,T . It is clear that the study of Ψε can follow exactly the lines of the study of ΨT , yielding a unique ﬁxed point if ε is small enough, the size of admissible ε being controlled by the L2 -norms of zT and the norm of ωT as a linear form on C0p−1 (Ω, Rk ). These norms can in turn be bounded by the L2 norms of z0 and the norm of ω0 , respectively, multiplied by a continuous function T vT |1,∞ , |ϕvT,0 |1,∞ ). Proving that this is uniformly bounded for T < T0 is of max(|ϕ0,T therefore suﬃcient to get the contradiction we aim for, that is, that the solution can be uniquely extended beyond T0 . T T So, everything relies on proving the uniform boundedness of ϕv0,T , ϕvT,0 , and their derivatives over Ω. By Lemmas 7 and 9, these quantities are bounded by functions of |vT |1,T so that we have to prove that these can be bounded uniformly in T . However, it suﬃces to use the facts that vT satisﬁes a geodesic equation and that geodesics travel at constant speed. More precisely, deﬁning, for t ≤ T < T0 ,

2 2 ψt = vtT B + σ 2 |zt |2 , we have (recall that this does not depend on T as soon as T ≥ t) ψt ≡ ψ(0) so that

T

v

≤ T ψ(0) 1,T for all T . It is well known that minimizing geodesics have constant speed, but we must check that this property remains true for all the solutions of (22). This is proved in the appendix and is stated, for further reference, in the next lemma. 2 2 Lemma 4. If (j, v, z) is a solution of system (22) on [0, T [, then |vt |B + σ 2 |zt |2 is constant with respect to time. To prove the continuity of the solution, let (v, j, z) and (v , j , z ) be two solutions of system (22) with initial conditions (ω0 , z0 ) and (ω0 , z0 ), respectively. Using, in particular, the computation leading from (28) to (29), it is not to diﬃcult to obtain the estimate

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | eC|v|1,T 2 C |v | 2 1,T . + C σ 2 |z0 |2 + |ω0 | |v − v |1,T e

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

37

As we just have shown, |v|1,T = T |v0 |B , and this is smaller (up to a universal multiplicative constant) than |ω0 | so that

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | eCT |ω0 | 2 2 2 + C σ |z0 |2 + |ω0 | |v − v |1,T eCT |ω0 | . Gronwall’s lemma now allows us to conclude that, for some constant C which may now depend on T, |ω0 | , |ω0 | , |z0 |2 , and |z0 |2 , (31)

σ 2

2 2

|vt − vt |B ≤ C |ω0 − ω0 | +

|z0 | − |z0 | . 2

9. Normal coordinates in H 1 . We now consider the question, which motivated this paper, of whether the previous construction could be used as an indexing device for characterizing the deformations and variations of an object relative to a prototype. Fix an image j0 ∈ H 1 (Ω, Rd ). The computationally simplest way to describe an image j in a neighborhood of j0 is by the diﬀerence j − j0 . However, one cannot be satisﬁed with this representation which takes no account of the metric we have placed on JW . Among local charts related to the metric, normal coordinates on a Riemannian manifold are radial ﬂattenings of this manifold onto its tangent space in the sense that radial lines in this space correspond to geodesics on the manifold. They provide a very eﬃcient linear representation of the manifold and of its metric. Existence of such coordinates is a standard theorem in ﬁnite dimensions, and our purpose is to check how much of this result remains valid in our inﬁnite dimensional framework. In the previous sections, another candidate for local coordinates has emerged, which turns out to be closely related (it is in fact dual) to normal coordinates. We have proved that, for a ﬁxed j0 ∈ H 1 (Ω, Rd ), one can associate to any z0 ∈ L2 (Ω, Rd ) a unique solution of system (21). We introduce the function Mj0 : L2 → L2 , which assigns to z0 ∈ L2 the “image” j1 , where jt is the solution of (21) at time t. The following theorem shows that Mj0 shares some features of local coordinates on JW . Theorem 8. Let BH 1 (0, ε) denote the open ball in H 1 (Ω, Rd ) containing all z0 ∈ H 1 (Ω, Rd ) such that |z0 |H 1 < ε. Then, for all j0 ∈ H 1 (Ω, Rd ), there exists ε > 0 such that Mj0 restricted to BH 1 (0, ε) is continuous and one-to-one onto its image, equipped with the L2 -topology. Proof of Theorem 8. Continuity of Mj0 : L2 (Ω, Rd ) → L2 (Ω, Rd ) is a consequence of Theorem 7, and it trivially implies the continuity of the restriction Mj0 : H 1 (Ω, Rd ) → L2 (Ω, Rd ) for the H 1 (Ω, Rd )-topology. We show that this map is one-to-one in a neighborhood of 0. We ﬁrst have the following lemma. z0 |H 1 ) ≤ 1. Denote v ˜ the Lemma 5. Let j0 , z0 , z˜0 ∈ H 1 (Ω, Rd ), with max(|z0 |H 1 , |˜ time-dependent vector ﬁeld along the solution of (21) with initial condition (j0 , z˜0 ). Then, there exist a constant C and a function ε which depend only on j0 such that, for t > 0,

˜

(Mj (t˜ z0 ) − Mj0 (tz0 )) ◦ ϕv0,t − t[σ 2 (˜ z0 − z0 ) + dj0 K(dj0∗ (˜ z0 − z0 ))] 2 0 ≤ Ct |˜ z0 − z0 |2 ε(t),

´ AND LAURENT YOUNES ALAIN TROUVE

38

and limt→0 ε(t) = 0. The proof of Lemma 5 is given in Appendix G. To prove Theorem 8, we ﬁrst remark that

2

σ (˜ z0 − z0 ) + dj0 K(dj0∗ (˜ z0 − z0 )) 2 ≥ σ 2 |˜ z0 − z0 |2 so that

C v ˜

2

(Mj (t˜ 1 − z ) − M (tz )) ◦ ϕ ≥ tσ |˜ z − z | ε(j , t) , 0 j0 0 0 0 2 0 0,t 2 0 σ2

and the lower bound is nonvanishing as soon as t is small enough. Remark that we have, for j1 , j2 ∈ H 1 (Ω, Rd ), the inequality d(j1 , j2 ) ≤

1 |j1 − j2 |2 σ

since the right-hand side is an upper bound of the length of the curve jt = (1−t)j1 +tj2 2 2 1 t (since choosing v ≡ 0 and σ 2 z ≡ j2 − j1 , we have wt (vt , zt ) ∈ ∂j |zt |2 = ∂t and σ 0 2 |j2 − j1 |2 /σ 2 ). So continuity of Mj0 for the d-topology on its image is also true. According to Lemma 5, normal coordinates (which are time derivatives at t = 0 of geodesics) are related to M by the relation (we use the standard exponential notation) expj0 (Sz0 ) = Mj0 (z0 ), where S is deﬁned by Sz σ 2 z + Dj0 K(Dj0∗ z). This indicates that a good approximation of the metric in terms of the z-coordinate would be 2

|z1 − z2 |j0 = z1 − z2 , S(z1 − z2 )2 , which satisﬁes |z1 |j0 = d(j0 , Mj0 (z1 )) in a neighborhood of 0. 10. Experiments. In this section, we propose a preliminary set of experiments to illustrate the information contained in the z-coordinate described above. Experiments in Figures 1, 2, and 3 were conducted in two steps: given two images j0 and j1 , we ﬁrst computed the minimizing geodesic between them, yielding a trajectory (jt , zt , vt ) and the corresponding ﬂow ϕvt . Then, using j0 again, and the obtained value z0 on the minimizing geodesic, we computed the solution of (21) until time t = 1. The obtained values (jt , zt , vt ) could then be compared with those which have been computed along the geodesics. In Figure 4, the initial j0 is the same as in Figures 2 and 3, but the z0 is the average of the two so that it does not correspond to any precomputed geodesic in the image space. The result is quite interesting, because it still possesses characteristics of a human face and can be compared to the result of a simple linear combination of the target images in Figures 2 and 3. The numerical implementation of both operations (minimization of the geodesic energy and integration of (21)) must be done with some care in order, in particular, to avoid instabilities due to the conservation part of the energy. Details will be provided in a forthcoming paper.

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

39

Fig. 1. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

Appendix A. Proofs of Propositions 2 and 3. A.1. Proof of Proposition 2. The proof relies on a sequence of standard measurability arguments, of which we sketch only the main steps. First let (wn )n∈N be a Hilbert basis of W . Since, for any u ∈ Cc∞ (Ω, Rd ) and w = (v, z) ∈ W , j → lj,u (w) (which has been deﬁned in (6) by σ 2 z, u2 + j, div(u ⊗ v)2 ) is continuous from L2 (Ω, Rd ) to R, the map j → wj,u lj,u (wn )wn n≥0

is measurable from L2 (Ω, Rd ) to W . By construction, we have, for w ∈ W , w , wju W = lj,u (wn )w , wn W = lj,u (w). n≥0

Thus, for γ ∈ Tj JW , we have p(γ) = Argmin |w|W : w , wj,u = γ , u for all u ∈ Cc∞ (Ω, Rd ) . Introducing a family (un )n∈N in Cc∞ (Ω, Rd ) which is dense in H01 (Ω, Rd ), the previous expression may be replaced by p(γ) = Argmin {|w|W : w , wj,un = γ , un for all n ≥ 0} .

´ AND LAURENT YOUNES ALAIN TROUVE

40

Fig. 2. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

For N ∈ N and λ > 0, we deﬁne (32)

pN,λ (γ) = Argmin

2 |w|W

+λ

N

w , wj,un W

2 − γ , un

.

n=0

N Clearly, we must have pN,λ (γ) = i=1 xi wj,ui , where ⎧

2

N 2 N N ⎨

x = Argmin

xn wj,un + λ xn wj,un , wj,un W − γ , ui

x ∈RN +1 ⎩

n=0

n=0

W

n =1

⎫ ⎬

+

1 2 |x | . ⎭ λ

For λ > 0, the optimal x is given by x = (A + I/λ)−1 y, where y ∈ RN +1 is such that yi = γ , ui and A is an (N + 1) × (N + 1) matrix with coeﬃcients given

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

41

Fig. 3. From top to bottom and from left to right: Initial image, target image, z-coordinate, reconstructed target image.

by an,n = wj,un , wj,un W . This implies that, if γt is a measurable path, the function t → pN,λ (γt ) is measurable. The measurability of p(γt ) is a consequence of the pointwise convergence of pN,N

(γt ) to p(γt ), which comes from the following argument: for all N and λ, we have pN,λ (γ) W ≤ |p(γ)|W , since the last term in (32) vanishes for w = p(γ). For the same reason, N n=0

2 1 pN,λ (γ) , wj,un − γ , un ≤ |p(γ)|W , λ

which implies that for all n, pN,N (γ) , wj,un → γ , un when N tends to inﬁnity. Moreover, for any weakly converging subsequence extracted from pN,N (γ) (which forms a weakly set in W ), we have, and denoting w∗ its limit,

N,N compact

∗

|w |W ≤ lim inf p (γ) W ≤ |p(γ)|W , and, for all n, w∗ , wj,un = γ , un by weak convergence, which is only possible when w∗ = p(γ). Hence t → p(γt ) is measurable if γt is measurable, and the proof of Proposition 2 is ended.

´ AND LAURENT YOUNES ALAIN TROUVE

42

Fig. 4. From top to bottom and from left to right: Initial image, target image, z-coordinate, obtained by averaging the z-coordinate in Figures 2 and 3, and obtained target image.

A.2. Proof of Proposition 3. We deduce from Proposition 2 that it is suﬃcient to prove the next proposition. Proposition 6. Let w ∈ L2 ([0, 1], W ) such that for any u ∈ Cc∞ (Ω×]0, 1[, Rd ) we have (33)

1

σ 2 zt , ut 2 + jt , div(ut ⊗ vt )2

dt = 0.

0

Then almost everywhere in t, wt ∈ Ejt . Proof. Let (un )n∈N be a family in Cc∞ (Ω, Rd ) dense in Cc∞ (Ω, Rd ) for the H 1 (Ω, Rd )norm. If we prove that for any n ∈ N, the function cn deﬁned by cn (t) σ 2 zt , un 2 + jt , div(un ⊗ vt )2 is vanishing almost everywhere, then by density, there exists a negligible set N such that for any t ∈ [0, 1] \ N and any u ∈ Cc∞ (Ω, Rd ) σ 2 zt , u2 + jt , div(u ⊗ vt )2 = 0,

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

43

which implies Proposition 6. Hence, let us consider n ∈ N. For any f ∈ Cc∞ ([0, 1], R), if u(t, x) f (t)un (x), we have from (33) that 1 cn (t)f (t)dt = 0 0

so that, by standard arguments, we get cn = 0 almost everywhere. Appendix B. Proof of Theorem 2. We start the (⇐) part in the case L2 ([0, 1], W ). Lemma 6. Let w = (z, v) ∈ L2 ([0, 1], W ). Let us deﬁne for any t ∈ [0, 1] t 2 jt j0 ◦ ϕt,0 + σ zs ◦ ϕt,s ds, 0

where ϕt is the ﬂow at time t associated with v. Then j is regular. Proof. Let us notice ﬁrst that t+h jt+h = jt ◦ ϕt+h,t + σ 2 (34) zs ◦ ϕt+h,s dt. t

From equality (34), the continuity in JW of j is straightforward. It is suﬃcient to prove that for any u ∈ Cc∞ (Ω×]0, 1[, Rd ), we have 1 1 2 ∂u jt , − dt = (35) σ zt , ut 2 + jt , div(ut ⊗ vt )2 dt. ∂t 2 0 0 Indeed, if (35) is proved, if for any t ∈ [0, 1] we denote γt (jt , wt ), we have for any u ∈ Cc∞ (Ω, Rd ), t → γt , u = σ 2 zt , u2 + jt , div(u ⊗ vt )2 measurable, and 1 1 |γt |jt ≤ |wt |W so that 0 |γt |2t dt ≤ 0 |wt |2W dt < +∞, and the lemma is proved. We have 1 1 1 jt+h − jt ∂u ut − ut−h , ut dt jt , jt , dt = − lim − dt = lim h→0 0 h→0 0 ∂t 2 h h 0 2 2 so that 1 t+h 1 1 ∂u 2 − jt ◦ ϕt+h,t − jt , ut 2 dt + σ jt , = lim zs ◦ ϕt+h,s , ut 2 ds dt. h→0 h 0 ∂t 2 0 t t+h However, jt ◦ ϕt+h,t − jt , ut 2 = t jt ◦ ϕs,t , div(ut ⊗ vs )2 ds so that 1 t+h 1 1 ∂u 2 jt , dt = lim jt ◦ ϕs,t , div(ut ⊗ vs )2 + σ zs ◦ ϕt+h,s , ut 2 ds dt − h→0 0 h ∂t 2 0 t

t+h 1 1 = lim jt , div(ut ⊗ vs ) ◦ ϕt,s |dϕt,s |2 + σ 2 zs , ut ◦ ϕs,t+h |dϕs,t+h |2 ds dt. h→0 0 h t Since jt is uniformly bounded on L2 and |ϕt,s − I|1,∞ = (|t − s|) (since B is continuously embedded in C 1 (Ω, Rk )), there exists C > 0 such that (36)

1

1 1 t+h

jt , div(ut ⊗ vs ) ◦ (ϕt,s |dϕt,s | − I)2 dsdt ≤ C(h) |vt |1,∞ dt

0 h t

0 1 1/2 2 ≤ C (h) (37) |wt |W dt . 0

´ AND LAURENT YOUNES ALAIN TROUVE

44

Now, using again the fact that jt is uniformly bounded in L2 and fact that C([0, 1], L2 (Ω, Rk )) is dense in L2 ([0, 1], L2 (Ω, Rk )), we get

1 1 t+h

jt , div(ut ⊗ vs ) − div(ut ⊗ vt )2 dsdt

(38)

0 h t 1 t+h 1 ≤C |div(ut ⊗ vs ) − div(ut ⊗ vt )|2 dsdt 0 h t → 0 when h → 0. At this point we have proved that 1 1 1 lim (39) jt ◦ ϕt+h,t − jt , ut 2 dt = jt , div(ut ⊗ vt )2 dt. h→0 h 0 0 Still using the fact that |ϕt,s − I|1,∞ = (|t − s|) and the fact that |ut |1,∞ is uniformly bounded, we have 1 t+h 1 1 |zs |2 ds (40) σ 2 zs , ut ◦ (ϕs,t+h |dϕs,t+h | − I)2 dsdt ≤ Cσ 2 (h) 0 h t 0 1 1/2 ≤ Cσ(h) (41) |wt |22 . 0

Finally, since |ut |∞ is uniformly bounded, we get

1 t+h

1 1 t+h

zs − zt , ut 2 dsdt ≤ lim C |zs − zt |2 dsdt = 0. lim

h→0 h→0 0 h t 0 t Hence the proof of the lemma is ended. Let us consider the (⇒) part of Theorem 2 for H 1 ([0, 1], JW ). Let j ∈ H 1 ([0, 1], JW ) be a regular path, and let wt = p( ∂j ∂t ) for any t ∈ [0, 1]. We get from Proposition 2 that w ∈ L2 ([0, 1], W ). Hence, let us deﬁne the new path j by t jt = j0 ◦ ϕt,0 + σ 2 zs ◦ ϕt,s ds, 0

where ϕ is the ﬂow associated with v. From the (⇐) part, we get that j is regular ∂j ∞ d ∞ and that ∂j ∂t = ∂t . Now let u0 ∈ Cc (Ω, R ). For any f ∈ Cc (]0, 1[, R) if u(t, x) = u0 (x)f (t) for any x ∈ Ω and t ∈ [0, 1], we have from the integration by parts formula for a regular path 1 r(t)f (t)dt = 0, 0

jt , u2 .

Since r is continuous and r(0) = 0, we get r ≡ 0. where r(t) = jt , u2 − Considering arbitrary u0 , we get ﬁnally jt = jt for any t ∈ [0, 1]. Since the (⇒) part for C 1 ([0, 1], JW ) is a straightforward consequence of the deﬁnition of C 1 curves and of the (⇒) part for H 1 ([0, 1], JW ), we consider the (⇐) part for w ∈ C([0, 1], W ). We get from the corresponding part for L2 ([0, 1], W ) that (35) is still true. For any f ∈ Cc∞ ([0, 1], R) and any u ∈ Cc∞ (Ω, Rd ) we have 1 t f (t)jt , u2 dt = f (t) σ 2 zt , ut 2 + jt , div(u ⊗ vt )2 dt. − 0

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

45

One easily checks that t → σ 2 zt , ut 2 + jt , div(u ⊗ vt )2 is continuous as well as t → jt , u2 so that, considering smooth approximates of step functions, we deduce that s 2 js , u2 = j0 , u2 + σ zt , ut 2 + jt , div(u ⊗ vt )2 dt, 0

and the result is proved. Appendix C. Regularity results for AT . In this section, we collect a few useful results on how the regularity of the diﬀeomorphism AT (v) = ϕvT may be related to the norm on B, provided this norm can in turn control a suﬃcient number of derivatives; the ﬁrst result deals with boundedness. In the following, we assume at least that B is continuously embedded in C01 (Ω, Rk ) so that AT is well deﬁned for all T . In this case, ϕvT (x)

T

vs (ϕvs (x))ds.

=x+ 0

If vs had p space derivatives for all s, a formal diﬀerentiation of this equality yields (42)

d

p

ϕvT

T

dp (vs ◦ ϕvs )ds.

p

= d id + 0

This can be proved rigorously from rather standard arguments in the study of ODEs and is stated in the next lemma, for which we provide a proof for completeness, because of the small complication due to the fact that we have only an L1 control with respect to the t variable, instead of the usual uniform one. Lemma 7. If p ≥ 1 and B is embedded in C0p (Ω, Rk ), then, for all v ∈ L1 ([0, T ], Ω), v ϕ is p times diﬀerentiable and, for all q ≤ p, ∂ q v d ϕt = dq (vt ◦ ϕvt ). ∂t Moreover, there exist constants C, C such that, for all v ∈ L1 ([0, T ], Ω), (43)

sup |ϕvs |p,∞ ≤ CeC

|v|1,T

.

s∈[0,T ]

Proof. For further reference, we ﬁrst state Gronwall’s lemma. Lemma 8 (Gronwall). Asume that α and β are two positive, continuous functions on the interval [0, c] and that w(t) ≤ α(t) +

t

β(s)w(s)ds. 0

Then, w(t) ≤ α(t) + 0

t

t β(u)du α(s)β(s)e s ds.

´ AND LAURENT YOUNES ALAIN TROUVE

46

The continuity of x → ϕv0,t (x) is a direct consequence of this lemma since, for x, y ∈ Ω,

t

v v v v

vs (ϕ0,s (x)) − vs (ϕ0,s (y)) ds

|ϕ0,t (x) − ϕ0,t (y)| = x − y + 0

t

vs 1,∞ |ϕv0,s (x) − ϕv0,s (y)|ds,

≤ |x − y| + 0

and Gronwall’s lemma implies |ϕv0,t (x) − ϕv0,t (y)| ≤ |x − y| exp(C |v|1,T ).

(44)

Assume p = 1 and pass now to the diﬀerential of ϕv0,t . Fix x ∈ Ω and introduce the linear diﬀerential equation, formally obtained in (42) for p = 1, ∂Wt = dϕv0,t (x) vt Wt ∂t

(45)

with initial condition W (0) = δ ∈ Rk . We skip the argument ensuring the existence and uniqueness of a solution of this equation on [0, 1] and proceed to identifying it as Wt = dx ϕv0,t δ. Denote aε (t) = ϕv0,t (x + εδ) − ϕv0,t (x) /ε − Wt . For α > 0, introduce μt (α) = max {|dx vt − dy vt | : x, y ∈ Ω, |x − y| ≤ α} . The function dx vt ∈ C01 (Ω) being uniformly continuous on the compact set Ω, we have limα→0 μt (α) = 0. We may write t 1 t aε (t) = vs (ϕv0,s (x + εδ)) − vs (ϕv0,s (x)) ds − dϕv0,s (x) vs Ws ds ε 0 0 t dϕv0,s (x) vs aε (s)ds = 0 1 t (vs (ϕv0,s (x + εδ)) − vs (ϕv0,s (x)) − εdϕv0,s (x) vs (ϕv0,s (x + εδ) − ϕv0,s (x)))ds. + ε 0 Since for all y, y ∈ Ω |vt (y ) − vt (y) − dy vt (y − y)| ≤ μs (|y − y|) |y − y| , we may write |aε (t)| ≤ 0

t

|vs |1,∞ |aε (s)| ds + C(v) |δ|

1

μs (εC(v) |δ|)ds 0

for some constant C(v) which depends only on v. The fact that aε (t) tends to 0 when ε → 0 now is a direct consequence of Gronwall’s lemma and of the fact that 1 lim μs (α)ds = 0, α→0

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

47

which is true by the dominated convergence theorem, since μs pointwise converges to 0 and μs (α) ≤ 2 |v|1,∞ . This proves Lemma 7 in the case p = 1. The rest of the proof is by induction: let q0 ≤ p, q0 > 1, and assume that the result is proved for all q < q0 : ∂ q v d ϕt = dq (vt ◦ ϕvt ). ∂t This implies that for δ1 , . . . , δq ∈ Rk , we may write ∂ q v (l) (l) dl vt (δ1 , . . . , δl ), d ϕt (δ1 , . . . , δq ) = dϕv0,t vt dq ϕvt (δ1 , . . . , δq ) + ∂t q

l=2

(l)

each vector δk being a linear combination (with universal coeﬃcients) of terms of the kind dl ϕv0,t (δi1 , . . . , δil ) with l ≤ q + 1 − l (this result on the diﬀerentials of the composition of two functions can be easily proved by induction). This is a linear equation in dq ϕvt (δ1 , . . . , δq ), which is valid for q = q0 − 1, and the proof of its validity for q0 follows exactly the same lines as for p = 1. This expression also shows (using Gronwall’s lemma) that |dq ϕvt |∞ may be bounded by an expression of the kind

|v|1,T C˜ |dϕvt |∞ , . . . , dq−1 ϕvt ∞ exp C |v|1,T , where C˜ is a polynomial, which in turn implies (43). The same estimate is true for (ϕv )−1 . Lemma 9. If p ≥ 1 and B is continuously embedded in C0p (Ω, Rk ), there exist constants C, C such that, for all v ∈ L1 ([0, T ], Ω),

sup (ϕvs )−1 p,∞ ≤ CeC |v|1,T . s∈[0,T ]

Lemma 9 is a consequence of Lemma 7 and of the fact that (ϕvt )−1 = ϕw t with ws = −vt−s on [0, t]. We now pass to suﬃcient conditions for Lipschitz continuity of AT . For this, let ξ v, v ∈ L1 ([0, T ], B). For ξ ∈ [0, 1], denote vξ = (1 − ξ)v + ξv and ϕξ = ϕv . Lemma 10. t ξ ξ ∂ vξ (46) dϕvξ (x) ϕvut (vu − vu ) ◦ ϕvs,u (x)du. ϕs,t (x) = s,u ∂ξ s Proof. Let us ﬁrst start with a formal diﬀerentiation of ξ

∂ϕvs,t ξ = vtξ ◦ ϕvs,t ∂t with respect to ξ, which yields ξ ∂ ∂ vξ d ξ ϕ = (vt − vt ) ◦ ϕvt + dϕvξ vtξ ϕvs,t , s,t ∂t ∂ξ s,t dξ

which naturally leads us to introduce the solution of the diﬀerential equation (47)

ξ ∂ Wt = (vt − vt ) ◦ ϕvt + dϕvξ vtξ Wt s,t ∂t

´ AND LAURENT YOUNES ALAIN TROUVE

48

with initial condition Ws = 0. Noting that we have already encountered this equation ξ without the constant term in (45), the solution of which is of the form dx ϕv0,t δ, a standard argument by variation of the constant shows that the solution of (47) is given by the right-hand term of (46). Therefore, the proof boils down to show that the interversion of derivatives underlying the formal argument above can be made rigorous. For this, it clearly suﬃces to consider the problem in the vicinity of ξ = 0. The proof in fact follows the same lines as the proof of Lemma 7: letting Wt be the solution of (47), we let ξ aξ (t) = ϕvs,t (x) − ϕvs,t (x) /ξ − Wt and express it under the form, letting hu = vu − vu , s

1 + ξ

t

aξ (t) =

t s

t

ξ

(hu (ϕvs,u (x)) − hu (ϕvs,u (x)))du

dϕvs,u vu aξ (u)du +

s

ξ vu (ϕvs,u (x)) − vu (ϕvs,u (x)) − ξdϕvs,u (x) vu ϕvs,u (x) − ϕvs,u (x) du. ξ

The proof can proceed exactly as that of Lemma 7, provided it has been shown that ξ |ϕvs,u (x)−ϕvs,u (x)| tends to 0 with ξ, which is again a direct consequence of Gronwall’s lemma and of the inequality t

ξ

ξ

t

v

|vu |1,∞ ϕvs,u (x) − ϕvs,u (x) du + ξ |hu |∞ du.

ϕs,t (x) − ϕvs,t (x) ≤ s

s

This lemma implies, in particular, that (48)

1

ϕvs,t (x) − ϕvs,t (x) = 0

s

t

d ϕv ξ

s,u (x)

ϕvut (vu − vu ) ◦ ϕvs,u (x)dudξ, ξ

ξ

which almost immediately leads to the following result (by computing diﬀerentials and applying Lemma 7). Lemma 11. Assume that B is continuously embedded in C0p (Ω). If v, v ∈ 1 L ([0, T ], B), we have, for t ≤ T ,

v

ϕt − ϕvt

Cp (|v|1,t +|v |

p−1,∞

≤ Cp |v − v |1,t e

1,t

)

for some constant Cp which depends only on p. √ The same results apply on L2 ([0, 1], B), since |v|1,T ≤ T |v|2,T , but, in this space, weak continuity is true under more general conditions. Theorem 9 (Trouv´e, Dupuis, et al). Assume that B is continuously embedded in C0p (Ω, Rk ). Then the map ˜ T : L2 ([0, T ], B) → C p ([0, T ] × Ω, Rk ), A v → ϕv. (.) is continuous for the weak topology on L2 ([0, 1], B) and the norm |.|T,p−1,∞ on C p ([0, T ]× Ω, Rk ) deﬁned by |ϕ|T,p−1,∞ = ess.sup(|ϕt |p−1,∞ , t ∈ [0, T ]).

49

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Moreover, assume that the embedding is compact, that vn converges weakly to v, and that there exists a constant A such that, for all n and almost all s ∈ [0, 1], |vsn |B ≤ A. Then, for all x ∈ Ω and t ∈ [0, T ], n

dpx ϕvt → dpx ϕvt . Recall that vn converges to v in the weak topology on L2 ([0, 1], B) if and only if, for all w ∈ L2 ([0, 1], B),

1

lim

n→∞

0

Ω

vn (t) , w(t)B dt =

0

1

Ω

v(t) , w(t)B dt.

Proof. The proof of this theorem, which is sketched here for completeness, relies on the remark that, since vn weakly converges, it is bounded in L2 ([0, 1], B), and Lemma 7 readily implies that (ϕvn ) and their space derivatives up to order p − 1 are equicontinuous sequences in space. Equicontinuity in time comes by applying the Cauchy–Schwarz inequality to t dq ϕvt − dq ϕvs = dq (vu ◦ ϕvu )du. s

Ascoli’s theorem implies compactness of (ϕvn ) for the |.|T,p−1,∞ -topology, and it remains to identify a limit of any converging subsequence as ϕv . Denoting this limit by ψ, one deduces from t n vn ϕ0,t (x) = vsn (ϕv0,s (x))ds 0 n

v to ϕt the fact that and the convergence of ϕ0,t

t

vsn (ψs (x))ds + o(n),

ψt (x) = 0

t and the conclusion comes after the remark that w → 0 ws (ψs (x))ds is a continuous linear functional on L2 ([0, 1], B) so that the weak convergence of vn to v implies that ψt (x) =

t

vs (ψs (x))ds 0

and ψt = ϕv0,t . We now prove the pointwise convergence of the pth derivative. We know that d p v d ϕ = dϕvt vdpx ϕvt + Qvt (x), dt x t where Qvt (x) depends on the derivatives of v evaluated at ϕvt (x) and on the p − 1 ﬁrst space derivatives of ϕvt . We may therefore write t t n n dϕvs n v(dpx ϕvs − dpx ϕvs )ds + (dϕvs v − dϕvs n vn )dpx ϕvs ds dpx ϕvt − dpx ϕvt = 0

0

t

n

(Qvs (x) − Qvs (x))ds.

+ 0

´ AND LAURENT YOUNES ALAIN TROUVE

50

t

n

The ﬁrst integral may be bounded by C(|vn |1,T ) 0 dpx ϕvs − dpx ϕvs ds, and the result will be a consequence of Gronwall’s lemma, provided we show that the remaining terms tend to 0. Consider the second integral, which may be written t t (dϕvs v − dϕvs vn )dpx ϕvs ds + (dϕvs vn − dϕvs n vn )dpx ϕvs ds. 0

0

The ﬁrst term tends to 0 because w → 0

t

dϕvs (x) wdpx ϕvs ds

is a continuous linear functional on L2 ([0, t], B) and vn weakly converges to v in this space. To estimate the second one, introduce, for A, ε > 0, the number C(A, ε) = max {|dx w − dy w| : x, , y ∈ Ω, |x − y| ≤ ε, |w|B ≤ A} . The compact embedding assumption implies that, A being ﬁxed, C(A, ε) tends to 0 when ε tends to 0. Using this notation, we have t t n n p v (dϕvs v − dϕvs n v )dx ϕs ds ≤ C (|vsn |B , |ϕvs − ϕvs n |∞ ) |dpx ϕvs |∞ ds 0 0 t C (A, |ϕvs − ϕvs n |∞ ) |v|B ds, ≤ 0

where A = ess.sup {|vsn |B , n ≥ 0, s ∈ [0, 1]}. The last upper bound now tends to 0, by dominated convergence. Finally, a generic term of Qvt being dkϕvt (x) vt (di1 ϕvt , . . . , dik ϕvt ), we can use the same argument to prove its pointwise convergence. Appendix D. Action of diﬀeomorphisms on images. The next theorem provides results concerning the regularity of the action of diﬀeomorphisms on L2 (Ω, Rd ) and H 1 (Ω, Rd ). Theorem 10. (i) Let ϕ be a diﬀeomorphism of Ω such that ϕ and ϕ−1 have uniformly bounded ﬁrst derivatives on Ω. Then, if i ∈ L2 (Ω, Rd ) (resp., i ∈ H 1 (Ω, Rd )), also i ◦ ϕ ∈ L2 (Ω, Rd ) (resp., i ◦ ϕ ∈ H 1 (Ω, Rd ) and dx (i ◦ ϕ) = dϕ(x) i.dx ϕ). (ii) Moreover, for all M > 0 and for all i ∈ L2 (Ω, Rd ), there exists a function εM (i, η) such that, for all ϕ, ϕ such that

−1

−1

max |ϕ|1,∞ , ϕ 1,∞ , |ϕ |1,∞ , ϕ

≤ M, 1,∞

we have |i ◦ ϕ − i ◦ ϕ|2 ≤ εM (i, |ϕ − ϕ |1,∞ ), and εM (i, η) → 0 when η → 0. The same statement is true for i ∈ H 1 (Ω, Rd ), the L2 (Ω, Rd )-norm being replaced by the H 1 (Ω, Rd )-norm.

51

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Proof. We start with (i) and give the proof for H 1 (Ω, Rd ), since it contains exactly the arguments which are valid for L2 (Ω, Rd ). Fix ϕ and let Lϕ be deﬁned by Lϕ (i) = i ◦ ϕ. The vector space Cd∞ = C ∞ (Ω, Rd ) of restrictions to Ω of inﬁnitely diﬀerentiable functions on Rk taking values in Rd is dense in H 1 (Ω, Rd ) [10]. The linear map Lϕ is continuous from Cd∞ (with the topology induced by H 1 (Ω, Rd )) to H 1 (Ω, Rd ); indeed, for i ∈ Cd∞ ,

2 2 |Lϕ (i)|H 1 = |i ◦ ϕ|22 + |dϕ i dϕ|22 ≤ |i|22 dϕ−1 ∞ + |di|22 |dϕ|2∞ |dϕ−1 |∞ ≤ C |i|H 1 since the ﬁrst derivatives of ϕ and ϕ−1 are bounded. Thus, Lϕ restricted to Cd∞ ˜ ϕ on H 1 (Ω, Rd ). If i ∈ H 1 (Ω, Rd ) and in can be extended to a continuous function L ∞ is a sequence of elements of Cd which converges to i when n tends to inﬁnity (so ˜ ϕ (i) in H 1 (Ω, Rd )), then, because convergence in H 1 (Ω, Rd ) implies that in ◦ ϕ → L convergence in L2 (Ω, Rd ), a subsequence of in can be extracted which converges almost ˜ n (i). If N ⊂ Ω everywhere to i and such that in ◦ ϕ converges almost everywhere to L −1 null

Lebesgue measure, then it is also the case for ϕ (N ) (by boundedness of

has −1 ˜ ϕ = Lϕ .

dϕ ), so that in ◦ ϕ also converges almost everywhere to i ◦ ϕ, yielding L 1 2 Now, since the map i → di is obviously continuous from H to L , so is i → d(Lϕ (i)). But, since this map coincides with i → dϕ i dϕ on Cd∞ , and this last map is also continuous on H 1 (Ω, Rd ) (by the previous computation), we get equality over all H 1 (Ω, Rd ), again by density of Cd∞ . For (ii), we ﬁrst consider the L2 (Ω, Rd ) case. Let i, ϕ , ϕ, and M be as in the theorem, and ﬁx s ∈ C ∞ (Ω, Rd ); we have |i ◦ ϕ − i ◦ ϕ|2 ≤ |i ◦ ϕ − s ◦ ϕ |2 + |s ◦ ϕ − s ◦ ϕ |2 + |i ◦ ϕ − s ◦ ϕ|2 . First notice that |i ◦ ϕ − s ◦ ϕ |2 = 2

−1

−1

dϕ |i − s|2 dx ≤ C ϕ

Ω

2

1,∞

|i − s|2

for some constant C. For the middle term, we have

|s ◦ ϕ − s ◦ ϕ |2 ≤

1

0

dϕ+t(ϕ −ϕ)s (ϕ − ϕ) dt 2

≤ |ϕ − ϕ|∞

0

1

dϕ+t(ϕ −ϕ)s dt 2

≤ C(M ) |ds|2 |ϕ − ϕ|∞ . We thus get |i ◦ ϕ − i ◦ ϕ|2 ≤ C(M ) (|i − s|2 + |ds|2 |ϕ − ϕ|∞ ) . Letting εM (i, η) C(M )

inf

s∈C ∞ (Ω)

(|i − s|2 + |ds|2 η)

yields the conclusion of the theorem in the L2 (Ω, Rd ) case, the H 1 (Ω, Rd ) case being handled similarly.

´ AND LAURENT YOUNES ALAIN TROUVE

52

Appendix E. Proof of Lemma 2. We must compute the derivative at ε = 0 of 1 U = 2 σ ε

2

j0 ◦ ϕv+εh − j1

1,0 dx. 1 −1 ds |dϕv+εh Ω 1,s | 0

First, we notice the equation σ 2 zt (x) =

(49)

j1 ◦ ϕvt,1 − j0 ◦ ϕvt,0 , 1 |dϕvt,s |−1 ds 0

which implies that (diﬀerentiating at ε = 0) 1 dU ε d d

v+εh

−1 2 2 = −2 z1 , dϕv1,0 j0 ϕv+εh dϕ |z − σ | , ds. 1 1,s dε dε 1,0 dε 0 2 2 Starting with the ﬁrst term and using Lemma 10, we have 1 d dϕv1,0 j0∗ z1 , dϕv1t ϕvt,0 ht ◦ ϕv1t dt z1 , dϕv1,0 j0 ϕv+εh = − 1,0 dε 2 0 2 1 =− dϕvt,0 j0∗ z1 ◦ ϕvt1 |dϕvt1 | , dϕvt,0 ht dt 0

1

=− 0

=−

1

(dϕvt,0 )∗ dϕvt,0 j0∗ zt , ht

2

dt 2

K (dϕvt,0 )∗ dϕvt,0 j0∗ zt , ht dt B

0

because of the identity zt = z1 ◦ ϕvt1 |dϕvt1 |. We now pass to the second term, for which we use the equality # t $

v+εh −1

dϕt,s = exp div(vu + εhu ) ◦ ϕv+εh du , t,u s

which is a consequence of Lemma 7 and standard computations on the derivative of the determinant. This implies that d

v+εh

−1

v

−1 t dϕt,s = dϕt,s div(hu ) ◦ ϕvt,u du dε s u

−1 t

+ dϕvt,s

dϕvt,u div(vu ) dϕvtτ ϕvτ u hτ ◦ ϕvtτ dτ du s t

v −1 t v div(hu ) ◦ ϕt,u du = dϕt,s

s

v −1 t τ

dϕvt,u div(vu )dϕvtτ ϕvτ u hτ ◦ ϕvtτ dudτ. − dϕt,s s

s

We may notice that

−1

∇ |dϕvτ s |

−1 , ξ = |dϕvτ s | s

τ

dϕvτ u (divvu )dϕvτ u (ξ)du

53

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

to identify the last term as

v −1 t

dϕv ϕvτ s ∇ϕv |dϕvτ s |−1 , hτ ◦ ϕvtτ dτ

dϕt,s

tτ tτ s

so that d

v+εh

−1

v

−1 t dϕt,s = dϕt,s div(hu ) ◦ ϕvt,u du dε s t −1 −1 ∇ϕvtτ |dϕvτ s | , hτ ◦ ϕvtτ dτ. |dϕvtτ | − s

Therefore, 1 1 1 d

v+εh

−1 2 2 −1 dϕ1,s |z1 | , |z1 | , |dϕv1s | div(hu ) ◦ ϕv1u duds ds = dε 2 0 0 s 2 1 1 2 −1 −1 − |z1 | , |dϕv1u | ∇ϕv1u |dϕvus | , hu ◦ ϕv1u duds

s

0

1

0 1

1

s 1

= − 0

−1 2 −1

|zu | , |dϕvu1 | dϕvu1 ϕv1s div(hu ) duds 2

−1

|zu | , ∇ |dϕvus |

2

, hu

s

duds. 2

Introducing quv

u

|dϕvus |−1 ds,

0

this may be written 1 1 d

v+εh

−1 2 2 dϕ1,s |z1 | , quv |zu | , div(hu ) du ds = dε 2 0 0 2 1 2 − |zu | , ∇quv , hu du 0

1

=− 0

1

−

2

2

K∇ (quv |zu | ) , hu 2

K(|zu | ∇quv ) , hu

0

Now, deﬁning functions 2

Ctv σ 2 qtv |zt |

(50) and (51)

Dtv σ 2 |zt | ∇qtv + 2[dϕvt,0 ]∗ dϕvt,0 j0∗ zt , 2

Proposition 5 implies dU ε = dε

0

1

ht , K.Dtv + K∇ Ctv B dt,

B

du

B

du.

2

´ AND LAURENT YOUNES ALAIN TROUVE

54

which is the conclusion of Lemma 2. Appendix F. Proof of Lemma 4. We prove that solutions of system (22) 2 2 travel at constant speed and therefore compute the derivative of |vt |B + σ 2 |zt |2 for such a solution. Starting with the second term, we have zt = z0 ◦ ϕvt,0 dϕvt,0 , which implies, after a change of variables,

−1 2 2

|zt |2 = |z0 |2 dϕv0,t dx. Ω

Using the identity

v

dϕs,t = exp

(52)

t

div(vu ) ◦

ϕvs,u du

,

s

we obtain d 2 |zt |2 = − dt

(53)

Ω

−1 2

|z0 |2 dϕv0,t div(vt ) ◦ ϕv0,t dx.

2

To study the variation of |vt |B , we start with the computation of the derivative of vt , w for a ﬁxed w ∈ B. Applying formula (28) for a solution of (22) yields

−1 v

σ 2 1 2

v

−1 vt , wB = |z0 | , ( dϕ0,s ∇ξs,t , λvt (w) ) ds div(w) ◦ ϕv0,t − dϕv0,t

2 0

v with ξs,t have

+ (ω0 , λvt (w))

= dϕv0,t / dϕv0,s and λvt (w) = (dϕv0,t )−1 w ◦ ϕv0,t . From formula (52), we v ξs,t

t

(divvu ) ◦

= exp

ϕv0u du

,

s

which implies that v dξs,t

=

v ξs,t s

so that

t

dϕv0u (divvu )dϕv0u du

σ 2 t 2

v

−1 |z0 | , dϕ0,s div(w) ◦ ϕv0,t ds vt , wB = (ω0 , λvt (w)) + 2 0 2 t t

−1 σ 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) duds − 2 0 s σ 2 t 2

v

−1 v = (ω0 , λt (w)) + |z0 | , dϕ0,s div(w) ◦ ϕv0,t ds 2 0 2 t u

−1 σ 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) dsdu. − 2 0 0

We now compute the time diﬀerential of each term which appears in this expresv d v sion. Denote λt (w) = dt λt (w). We have v

λt (w) =

d (dϕv0,t )−1 w ◦ ϕ0,t = −(dϕv0,t )−1 dϕv0,t vt w ◦ ϕ0,t + (dϕv0,t )−1 dϕv0,t wvt ◦ ϕ0,t . dt

55

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

Next, we have

−1 d t 2

v

−1 2

|z0 | , dϕ0,s div(w) ◦ ϕv0,t ds = |z0 | , dϕv0,t div(w) ◦ ϕv0,t dt 0 t

−1 2

|z0 | , dϕv0,s ∇ϕv0,t (div(w))vt ◦ ϕv0,t ds + 0

and d dt

t 0

u

0

t

= 0

−1 2

|z0 | , dϕv0,s dϕv0,t (divvt )dϕv0,t λvt (w) dsdu

t + 0

−1 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λvt (w) dsdu

0

u

−1 v 2

|z0 | , dϕv0,s dϕv0u (divvu )dϕv0u λt (w) dsdu.

Putting everything together, we have σ2

−1 d v 2

|z0 | , dϕv0,t div(w) ◦ ϕv0,t vt , wB = ω0 , λt (w) + dt 2 σ 2 t 2

v

−1 + |z0 | , dϕ0,s dϕv0,t (divw)vt ◦ ϕv0,t dsdu 2 0 σ 2 t 2

v

−1 − |z0 | , dϕ0,s dϕv0,t (divvt )dϕv0,t λvt (w) dsdu 2 0 σ 2 t u 2

v

−1 v |z0 | , dϕ0,s dϕv0u (divvu )dϕv0u λt (w) dsdu. − 2 0 0 v

d A little care must be taken in writing, as we did above, dt (ω0 , λvt (w)) = (ω0 λt (w)), v v v since this requires proving that (λt+ε (w) − λt (w))/ε converges to λt (w) for the (p − 1, ∞)-norm. This is indeed true in our case, because of the fact that w ∈ B allows us to control the uniform norm of its diﬀerentials up to order p, and the diﬀerentials of ϕvt up to the same order are solutions of a linear diﬀerential equation which ensures their uniform continuity. We now use the identity (which is justiﬁed below)

d 2 |vt |B = 2 lim vt+ε − vt , vt B /ε, ε→0 dt

(54)

2

which implies that, to compute the time diﬀerential of |vt |B , it suﬃces to use the obtained expression for the derivative of vt , wB with w = vt and multiply it by v 2. Since λt (vt ) = 0, and because of (53), we see that all terms cancel, yielding 2 2 d 2 dt (|vt |B + σ |zt |2 ) = 0. To show (54), one writes 2

2

2

(|vt |B − |vt |B − vt+ε − vt , vt B )/ε = |vt+ε − vt |B /ε, and the result is obtained by proving that, for w ∈ B, |vt+ε − vt , wB | = O(ε) |w|B , which can be done by a direct estimation of

d dt vt

, wB .

´ AND LAURENT YOUNES ALAIN TROUVE

56

Appendix G. Proof of Lemma 5. It suﬃces to prove this result for smooth z0 , z˜0 , j0 . It is straightforward that Mj0 (tz0 ) = jt , where j is the solution of (21) with initial conditions (j0 , z0 ). Let ˜jt = Mj0 (t˜ z0 ). Introduce also the corresponding (vt , zt ) and (˜ vt , ˜zt ). Introduce the notation η = ˜j − j, ζ = ˜z − z, and α = v ˜ − v. Since we have assumed smooth trajectories, we may write ∂jt = σ 2 zt − djt vt ∂t and ∂zt = −div(zt ⊗ vt ) ∂t and similar equations for the trajectory with initial condition (j0 , z˜0 ). Computing the diﬀerences along both trajectories yields ⎧ ∂ηt ˜t = σ 2 ζt − djt αt , ⎨ ∂t + dηt v (55) ⎩ ∂ζt ˜t ) = −div(zt ⊗ αt ). ∂t + div(ζt ⊗ v Since & v˜ ∂ζt ∂ %

v˜

˜ ˜ dϕ0,t ζt ◦ ϕv0,t + div(ζt ⊗ v = dϕ0,t

˜t ) ◦ ϕv0,t ∂t ∂t

˜

˜

div(zt ⊗ αt ) ◦ ϕv0,t = − dϕv0,t , the second term yields ζs ◦

˜ ϕv0,s

˜ −1

(˜ = dϕv0,s z0 − z 0 ) −

s

v˜

˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕv0,s ,

0

and the ﬁrst one implies ηt ◦

˜ ϕv0,t

=σ

t

ζs ◦

2

˜ ϕv0,s ds

0

−

t ˜ [djs αs ] ◦ ϕv0,s ds.

0

Replacing ζ in the last equation gives (56) ˜ ηt ◦ ϕv0,t = t[σ 2 (˜ z0 (.) − z0 (.)) − dj0 (˜ v0 − v0 )] t s

v˜

2 v ˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕs,u du ◦ ϕv0,s ds −σ −

0 t

0

˜ [djs αs ] ◦ ϕv0,s − dj0 (˜ v0 − v0 ) ds +

0

t

v˜ −1

dϕ0,s − 1 (˜ z0 − z0 )ds 0

so that Lemma 5 reduces to evaluating the L2 -norm of the last three integrals. We shall use the fact that, for a function f ∈ L2 ([0, 1] × Ω, Rd ),

t

t

fs ds ≤ |fs |2 ds.

0

2

0

LOCAL GEOMETRY OF DEFORMABLE TEMPLATES

57

For t ∈ [0, 1], we also have, from (31), with ω0 = dj0∗ z0 and ω0 = dj0∗ z0 (here and in the following, we denote by const any quantity which depends only on j0 , z0 and z˜0 ), |αt |B ≤ const |z0 − z˜0 |2 .

(57)

This implies that

t s

v˜

˜ v ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕ0,s ds

0 0 2 t s

v ˜

v ˜

˜ ϕ ≤

dϕv0,s s,u div(zu ⊗ αu ) ◦ ϕ0,u duds 2 0 0 t s

v˜ −1

div(zu ⊗ αu ) ds =

dϕ0,s 2 0 0 t s |div(zu ⊗ αu )|2 duds ≤ const 0 0 t s |zu |H 1 |αu |B duds. ≤ const 0

0

v

dϕt,0 implies that |zu | 1 ≤ const |z0 | 1 so that The relation zt = z0 ◦ H H

t s

v˜

˜ ˜

dϕs,u div(zu ⊗ αu ) ◦ ϕvs,u du ◦ ϕv0,s ds

≤ const t2 |˜ z0 − z0 |2 .

ϕvt,0

0

0

2

A similar estimate is valid for the last integral in (56), since ||dϕv0,s |−1 −1|∞ ≤ const s. We ﬁnally consider the second integral in this equation. Since s js = j0 ◦ ϕvs,0 + σ 2 z0 ◦ ϕvs,0 |dϕvu0 | ◦ ϕvs,u du, 0

we have, letting γs =

ϕvs,0

◦

˜ ϕv0,s ,

v ˜ v v ˜ ˜ js α ◦ ϕ ˜ ϕ dϕv0,s 0,s = dγs j0 dϕv s,0 α ◦ ϕ0,s + Rs , 0,s

z0 − z0 |2 . We need to estimate and it is easy to check that |Rt |2 ≤ const t |z0 |H 1 |˜ t (58) 0

v v ˜ ˜ ϕ dγs j0 dϕv0,s ds α ◦ ϕ − dj α s 0 0 s,0 0,s t t v v ˜ ˜ ϕ dγs j0 (dϕv0,s = α ◦ ϕ − α ) ds + (dγs j0 α0 − dj0 α0 ) ds. 0 s,0 s 0,s 0

0

Start with the ﬁrst term, for which we must bound, for the L∞ -norm, the diﬀerence v v ˜ ˜ ϕ dϕv0,s s,0 α ◦ ϕ0,s − α0 or, equivalently, ˜ dϕvs,0 αs − α0 ◦ ϕvs,0 .

It is simple to check, from (57) and estimates we have used several times on the ˜ variations of the diﬀeomorphisms, that (dϕvs,0 − I)αs and α0 ◦ ϕvs,0 − α0 are bounded by const s |˜ z0 − z0 |2 . We now proceed to an upper bound for αs − α0 , for which we need to return to the expression obtained in (28), which yields σ2 s 2 −1 vs − v0 , wB = |z0 | , (|dϕv0u | div(w) ◦ ϕv0,s 2 0

−1 v − dϕv0,s ∇ξus , λvs (w)) du + z0 , dj0 (λvs (w) − w)2

´ AND LAURENT YOUNES ALAIN TROUVE

58 so that

˜ −1 v˜ σ 2 s 2

v˜

−1 ˜

αs − α0 , wB = |˜ z0 | , ( dϕ0u div(w) ◦ ϕv0,s − dϕv0,s ∇ξus , λvs˜ (w) ) du 2 0

−1

σ2 s 2 −1 v − |z0 | , (|dϕv0u | div(w) ◦ ϕv0,s − dϕv0,s ∇ξus , λvs (w)) du 2 0 + z˜0 , dj0 (λvs˜ (w) − w) 2 − z0 , dj0 (λvs (w) − w)2 . The diﬀerence of the ﬁrst two integrals takes the form σ 2 s ˜ z˜0 , Qvus (59) (w) − z0 , Qvus (w) du 2 0

−1

−1 v with Qvus (w) = |dϕv0u | div(w) ◦ ϕv0,s − dϕ v0,s ∇ξus , λvs (w). From Lemmas 7

˜ and 11, and from (57), we obtain the fact that Qvus (w) − Qvus (w) ≤ const |˜ z0 − z0 |2 |w|B so that the quantity in (59) is bounded by const s |˜ z0 − z0 |2 . Writing z˜0 , dj0 (λvs˜ (w) − w) 2 − z0 , dj0 (λvs (w) − w)2 = z˜0 − z0 , dj0 (λvs˜ (w) − w) 2 + z0 , dj0 (λvs˜ (w) − λvs (w)) 2

and using λvs˜ (w) − w ∞ ≤ const s (which is deduced from Lemma 7 and a com

putation of the diﬀerential of λvs˜ (w) with respect to s) and λvs˜ (w) − λvs (w) ∞ ≤ const s |˜ z0 − z0 |2 (from Lemma 11 and (57)), we ﬁnally conclude that

v v ˜ ˜ ϕ α ◦ ϕ ≤ const s |˜ z0 − z0 |2 , − α

dϕv0,s 0

s,0 0,s ∞

which implies that the ﬁrst integral in the right-hand term of (58) is bounded by const t2 |˜ z0 − z0 |2 . Consider now the last term of (58), namely, t (dj0 ◦ γs − dj0 ) α0 ds. 0

Since |α0 |∞ ≤ C |˜ z0 − z0 |2 , we must estimate |dγs j0 − dj0 |2 . By Theorem 10, this is a function of the kind

εM (dj0 , |γs − Id|∞ ) = εM (dj0 , ϕv˜ (s) − ϕv (s) ∞ ), ˜ where M depends only on |j0 |H 1 , |z0 |2 , |˜ z0 |2 . Since |ϕv0,s − ϕv0,s |∞ = O(s), we get (with another function ε) t (dj0 ◦ γs − dj0 ) α0 ds ≤ ε(j0 , t)t |˜ z0 − z0 |2 . 0

We need ﬁnally to consider the last line of (56) which can be easily bounded from above by ε(j0 , t)t |˜ z0 − z0 |2 . We now can collect the estimates we have obtained to conclude the proof of Lemma 5. REFERENCES [1] Y. Amit, U. Grenander, and M. Piccioni, Structural image restoration through deformable templates, J. Amer. Statist. Assoc., 86 (1989), pp. 376–387. [2] Y. Amit and P. Piccioni, A non-homogeneous Markov process for the estimation of Gaussian random ﬁelds with nonlinear observations, Ann. Probab., 19 (1991), pp. 1664–1678.

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